ELEMENTS 


OF 


ASTRONOMY, 


WITH 


Numerous  Examples  anti  Examination  papers 


BY 

GEORGE   W.   PARKER,   M.A. 


OF     TRINIT-.y    COLLEGE,     DUBLIN 


FOURTH     EDITION 


LONGMANS,    QKEEN,    AND    CO 

39,    PATERNOSTER    ROW,    LONDON 
NEW  YORK  AND  BOMBAY 

1902 


A3 


PRINTED  AT  THE 


BY  PONSONBY  &  WELDRICK. 


PREFACE. 


HPHE  present  volume  is  intended  to  meet  the  wants 
of  those  Students  whose  knowledge  of  Mathe- 
matics is  limited  to  an  acquaintance  with  the 
Elements  of  Euclid,  Algebra,  and  Plane  Trigono- 
metry. In  a  few  cases  easy  formulae  in  Dynamics 
are  introduced,  but  the  articles  containing  these 
may,  if  necessary,  be  omitted  without  a  breach  in 
the  continuity  of  the  work. 

Many  of  the  examples  have  been  selected  from 
papers  set  to  third  and  fourth  year  Students  of 
Trinity  College,  Dublin ;  while  a  considerable  num- 
ber have  been  chosen  with  a  view  to  assist  those 
reading  for  Degrees  in  the  London  and  Royal 
Universities. 

The  book  forms,  to  some  extent,  a  connecting 
link  between  the  many  popular  works  on  Astronomy 
and  more  advanced  treatises  on  the  subject.  The 
author,  therefore,  hopes  that  it  may  be  found  useful, 
not  only  by  those  for  whom  it  has  been  specially 


179755 


VI  PREFACE. 

written,  but  also  by  many  others  among  the  general 
public. 

The  author  is  much  indebted  to  MR.  PIERS 
WARD,  M.A.,  LL.B.,  for  his  kind  assistance  in  read- 
ing the  proof-sheets. 

13,  TBINITY  COLLEGE,  DUBLIN,  , 
July  19th,  1894. 


PREFACE 

TO  THE    SECOND   EDITION. 

IN  this  Second  Edition  the  work  has  been  carefully 
revised  and  some  new  matter  added.  The  number  of 
the  examples  has  also  been  increased.  The  Author 
desires  to  express  his  thanks  to  several  friends  who 
have  suggested  many  of  the  additions,  and  more 
especially  to  MR.  RAYMOND  WALTER  A.  SMITH  (Sch.), 
B.A.,  to  whom  most  of  the  corrections  are  due. 

13,  TRINITY  COLLEGE,  DUBLIN, 
March  13th,  1900. 


CONTENTS. 

CHAPTER  I. 

PROPERTIES   OF   THE   SPHERE.      DEFINITIONS. 

PAGE 

Fundamental  Definitions.     Apparent  diurnal  Motion  of  the  Heavens.     The 

Ptolemaic  and  Copernican  Systems.     Changes  in  the  Sun's  Decimation,        1 

CHAPTER  II. 

THE    EARTH. 

Altitude  of  the  Celestial  Pole.  Length  of  a  Degree  of  Latitude.  Magnitude 
of  the  Earth.  Proofs  of  the  Earth's  Rotation.  Foucault's  Experiment,  1 

CHAPTER  III. 

THE     OBSERVATORY. 

The  Transit  Instrument.  Various  Errors  «nd  Adjustments.  The  Meridian 
Circle.  Regulation  of  the  Clock.  Equatorial  and  Micrometer.  Alt- 
Azimuth  Instrument,  .  33 

CHAPTER  IV. 

ATMOSPHERIC   REFRACTION. 

Effect  on  the  Apparent  Position  of  a  Heavenly  Body.  Law  of  Refraction. 
The  Constant  Coefficient  determined  hy  various  methods.  Oval  appear- 
ances of  Sun  and  Moon, 53 


Vlil  CONTENTS. 

CHAPTER  V. 

THE   SUN. 

PAGE 

The  Sun's  apparent  Annual  Path.  Proofs  of  the  Earth's  Annual  Motion. 
The  Seasons.  Heat  from  the  Sun.  Sun  Spots.  Eotation  of  Sun. 
Twilight.  Its  duration  determined, 62 

CHAPTER  VI. 

THE  MOTIONS  OF  THE  PLANETS.       THE  SOLAE  SYSTEM. 

Definitions.  Phases  and  Brightness  of  the  Planets.  Periodic  Times  deter- 
mined. Kepler's  Laws.  Direct  and  Retrograde  Motions.  Rotations  of 
Planets.  Transits  of  Venus  and  Mercury.  Comets  and  Meteoric 
Showers.  Lengths  of  the  Seasons.  Eccentricity  of  Ellipse,  .  .  78 

CHAPTER  VII. 

PARALLAX. 

Law  of  Diurnal  Parallax.  Effect  on  the  apparent  position  of  a  body. 
Horizontal  Parallax  of  the  Moon  or  a  Planet  determined.  Finding  the 
Sun's  Parallax.  Magnitudes  of  Moon,  Sun,  and  Planets.  Annual 
Parallax — Bessel's  Method.  Annual  Parallax  of  Jupiter,  .  .  .114 

CHAPTER  VIII. 

DETERMINATION  OF  THE  FIRST  POINT  OF  AEIES.       PRECESSION,  NUTATION, 
AND   ABERRATION. 

Flamsteed's  Method  for  finding  the  Right  Ascension  of  a  Star.  Precession 
of  the  Equinoxes— its  Period  and  Physical  Cause.  Nutation.  Velocity 
of  Light.  Law  of  Aberration.  General  Effect  of  Aberration,  .  .  134 

CHAPTER  IX. 

THE   MOON. 

The  Moon's  Phases.  Determination  of  her  Synodic  Time  and  Sidereal  Period. 
Metonic  Cycle.  The  Moon  Rotates  round  an  axis.  Librations.  Harvest 
Moon  explained  geometrically.  Revolution  of  the  Moon's  Nodes. 
Height  of  a  Lunar  Mountain.  Physical  State  of  the  Moon,  .  .  149 


CONTENTS.  IX 

CHAPTER  X. 

ECLIPSES. 

PAGE 

Causes  of  Lunar  Eclipses.  Breadth  of  the  Earth's  Shadow  at  the  Moon. 
Solar  Eclipses.  Conditions  for  Eclipses.  Ecliptic  Limits.  Saros  of 
the  Chaldeans, 169 

CHAPTER  XI. 

TIME. 

The  Equation  of  Time.  Its  Causes.  Unequal  Lengths  of  Morning  and 
Afternoon.  Local  Time.  Mean  and  Sidereal  Time.  Reduction  of  Time. 
The  Julian  and  Gregorian  Calendars.  The  Sun-Dial,  ....  184 

CHAPTER  XII. 

APPLICATION   TO    NAVIGATION. 

Hadley's  Sextant.  Latitude  determined  by  various  Methods.  Mean  Local 
Time  calculated.  Longitude  by  Chronometers.  Lunar  Method  of 
determining  the  Longitude, 202 

CHAPTER  XIII. 

THE  FIXED  STAJBS.       SPECTRUM  ANALYSIS. 

Star  Magnitudes.  Clusters  and  Nebulae.  Proper  Motions.  Double,  Binary, 
and  Variable  Stars.  The  Solar  Spectrum.  Surface  of  the  Sun  and 
Solar  Prominences.  Spectra  of  Stars  and  Nebulae.  .  .  .  .214 

CHAPTER  XIV. 

MASSES   OF   THE   HEAVENLY   BODIES. 

The  Mass  of  the  Earth — Maskelyne's  Method  and  the  Cavendish  Experiment. 
Masses  of  the  Sun  and  Planets.  Masses  of  Binary  Stars.  Note  on  the 
Celestial  Globe, 227 


EXAMINATION  PAPERS  AND  MISCELLANEOUS  QUESTIONS,      ....     [l] 

INDEX, [17], 

b 


CHAPTER   I. 

PROPERTIES   OF   THE    SPHERE.       DEFINITIONS. 

1.  Definition. — A  SPHERE  is  a  solid  bounded  by  one 
surface,  such  that  all  right  lines  drawn  to  that  surface  from 
a  certain  point  within  it  are  equal.     That  point  is  called 
the  centre. 

A  sphere  might  also  be  defined  as  being  generated  by 
causing  a  circle  to  revolve  round  one  of  its  diameters.  Thus 
if  a  circular  hoop  or  ring  be  taken,  and  if,  having  fixed  two 
diametrically  opposite  points  in  its  circumference,  we  make  it 
revolve  round  the  diameter  joining  those  points,  we  see  that 
the  circumference  will  trace  out  the  surface  of  a  sphere. 

2.  Every  plane  meeting  a  sphere  cuts  its   surface  in  a 
circle. 

For,  let  DEF  represent  a  plane  section  of  a  sphere. 

From  0.  the  centre  of  the 
sphere,  let  fall  00'  perpendi- 
cular to  the  plane  DEF\  take 
any  point  P  in  the  circum- 
ference DEF.  Now,  since  00 
is  perpendicular  to  the  plane 
DEF,  it  must  be  perpendicular 
to  OP  which  lies  in  that  plane  ; 
therefore  (EUCLID,  i.  47), 


But  R  is  of  constant  length,  being  the  radius  of  the  sphere, 
and  p  is  constant ;  therefore  r  is  of  constant  length,  and 


PROPERTIES    OF    THE    SPHERE.  [CHAP.  I. 


therefore  as  P  changes  its  position  along  the  curve  LEF,  its 
distance  from  0'  is  constant.  Therefore  DEFm\Lst  be  a  circle. 
The  reader  can  illustrate  this  experimentally  by  taking 
an  apple  as  nearly  spherical  as  possible,  and,  with  a  knife, 
cutting  a  section  through.  On  examining  the  interior  of  the 
apple  thus  brought  to  view,  he  will  find  that  the  shape  of  the 
section  is  circular. 

3.  Definition. — A  great  circle  on  the  surface  of  a  sphere 
is  that  whose  plane  passes  through  the  centre  of  the  sphere. 
Thus  the  circle  AmnB*  (fig.  2)  is  a  great  circle. 

A  small  circle  is  such  that  its  plane  does  not  pass  through 
the  centre  of  the  sphere.  Thus  (fig.  1)  DBF  is  a  small 
circle. 

It  is  evident  that  all  great  circles  on  a  sphere  are  equal 
in  magnitude,  but  small  circles  are  not,  as  they  vary  from 
being  almost  great  circles  till  they  dwindle  down  to  mere 
points. 

Definition. — If  at  the  centre  of  a  circle  on  a  sphere  a 
perpendicular  be  erected  to  its  plane  and  produced  out  both 
ways,  the  two  points  in  which  it  cuts  the  sphere  are  called  the 
poles  of  that  circle. 

Thus,  if  at  the  centre  0  we 
erect  PQ  perpendicular  to  the 
plane  of  the  great  circle  AB, 
the  two  points  P,  Q  are  the 
poles  of  the  great  circle  AB. 

Definition.— Great  circles  A 
which  pass  through  the  poles 
of  another  great  circle  are 
called  secondaries  to  that  great 
circle.  Thus,  if  through  P 
and  Q  (fig.  2)  great  circles 
PmQ9  PnQ  be  drawn,  these  FIG  2. 

circles  are  secondaries  to  the  great  circle  AB. 


CHAP.  I.]  DEFINITIONS.  3 

N.B.—A.  great  circle  and  its  secondaries  cut  at  right 
angles.  Also  the  arcs  of  secondaries  Pm  and  Pn  drawn  from 
P  to  AB  are  equal,  and  each  is  equal  to  a  quadrant  =  90°. 

Apparent  Diurnal  Motion  of  the  Heavens.     Celestial  Sphere. 

4.  To  an  observer  situated  in  the  middle  of  a  level  plane 
the  appearance  which  the  heavens  present  to  him  is  that  of  a 
vast  hollow  dome  of  hemispherical  shape,  whose  base  appears 
to  rest  on  the  plane  on  which  he  stands,  meeting  it  in  a  circle. 
This  circle  which  bounds  his  view  of  the  heavens  is  called  the 
sensible  or  apparent  horizon,  the  plane  in  which  he  stands  being 
the  plane  of  the  sensible  horizon. 

If  now  the  time  of  observation  be  a  cloudless  night,  there 
will  be  seen,  apparently  spangled  over  the  concave  surface  of 
this  hollow  dome,  a  great  number  of  shining  bodies  or  specks 
of  light  called  stars. 

By  far  the  greater  number  of  these  bodies  always  main- 
tain nearly  the  same  situation  with  respect  to  each  other ; 
that  is,  if  the  angle  which  any  two  subtend  at  the  eye  of 
an  observer  be  measured,  it  is  found  never  to  undergo  any 
alteration  in  magnitude,  except  such  changes  as  are  of  so 
minute  a  character  as  only  to  be  observable  after  considerable 
intervals  of  time.  These  bodies  are  therefore  called  fixed  stars. 

To  even  the  most  careless  observer  it  will  at  once  appear 
that  all  these  fixed  stars  appear  to  move  across  the  sky  in  a 
common  direction  without  altering  their  positions  relative  to 
one  another.  Some  rise  in  the  eastern  horizon,  ascend  in  the 
sky,  and,  after  describing  arcs  which  seem  circular,  they  sink 
below  the  western  horizon,  only  to  reappear  again  in  the 
east  on  the  next  night  at  the  same  point  as  before,  the  whole 
revolution  of  each  being  completed  in  about  23h  56m  4s. 

Many  others  make  smaller  circuits  in  the  sky  such  that 
they  never  reach  the  horizon,  and  can  therefore  be  seen 
throughout  their  whole  course.  The  paths  of  these  stars  also 
seem  circular,  the  time  taken  by  each  to  complete  a  revolution 

B  2 


4  PROPERTIES   OF   THE   SPHERE.  [CHAP.  I, 

being  about  23h  56m  4s,  the  same  as  for  those  which  rise 
and  set.  Moreover,  their  apparent  diurnal  paths  all  seem  to 
be  round  one  common  point  as  pole.  That  point  is  called 
the  celestial  pole.  Those  stars  which  circle  so  closely  to  the 
pole  that  they  do  not  rise  and  set  are  called  circumpolar 
stars. 

In  order  to  more  accurately  observe  this  apparent  diurnal 
motion  of  the  heavenly  bodies,  it  is  necessary  to  have  a  tele- 
scope suitably  mounted,  called  an  equatorial,  such  as  is  to  be 
found  in  every  observatory.  This  telescope,  of  which  a  full 
description  will  be  given  later  on,  can  be  directed  to  any 
part  of  the  heavens,  so  that  any  star  desired  may  be  brought 
into  the  field  of  view.  Moreover,  by  means  of  a  clockwork 
arrangement,  the  field  of  view  can  be  made  to  move  uni- 
formly round  the  celestial  pole  in  the  same  direction  as  the 
stars  appear  to  move,  completing  its  revolution  in  23h  56m  4s. 
It  is  now  found  that  whatever  star  is  chosen,  once  the 
revolving  clockwork  apparatus  is  set  going,  it  is  possible  to 
keep  it  in  the  field  of  view  throughout  its  whole  course  above 
the  horizon.  If  the  reader  bear  in  mind  that  every  point  in 
the  field  of  view  of  the  telescope  moves  uniformly  in  a  small 
circle  round  the  celestial  pole,  and  that  the  time  of  complet- 
ing a  revolution  is  23h  56m  4s,  he  will  at  once  arrive  at  the 
following  conclusions : — 

(1)  The  stars  appear  to  move  in  small  circles  round  the 
celestial  pole. 

(2)  This  apparent  motion  is  uniform. 

(3)  The  time  occupied  in  completing  one  revolution  is 
the  same  for  all,  viz.  23h  56m  4s. 

5.  Celestial  Sphere. — If  two  observers  be  situated  at 
diametrically  opposite  points  of  the  earth,  say  in  England 
and  somewhere  about  New  Zealand,  each  will  see  an  appa- 
rently concave  hemispherical  surface  of  the  heavens  having  a 
celestial  pole.  If,  therefore,  the  observer  could  see 


CHAP.  I.J  DEFINITIONS.  5 

Leavens  at  a  glance,  the  appearance  would  be  that  of  a  com- 
plete sphere,  on  the  concave  surface  of  which  all  the  heavenly 
bodies  would  seem  to  be  situated,  and  revolving  round  two 
diametrically  opposite  points,  the  north  and  south  celestial 
poles.  This  apparently  spherical  surface  of  the  heavens  is 
called  the  celestial  sphere. 

It  is  usual,  from  a  mathematical  point  of  view,  to  regard 
the  celestial  sphere  as  a  sphere  of  infinitely  great  radius 
compared  with  any  distance  on  the  earth,  so  that  when  we 
say  that  the  earth  occupies  a  position  in  the  centre  of  this 
imaginary  sphere,  we  mean  that  we  may  regard  it  as  a 
mathematical  point,  i.e.  of  no  dimensions. 

The  Sim,  Moon,  and  Planets. 

6.  The  Sun. — Let  us  now  change  the  time  of  observation 
to  the  day-time,  and  see  if  the  sun  shares  in  the  diurnal 
revolution  common  to  the  fixed  stars.  At  first  sight  we 
might  come  to  the  conclusion,  that  his  apparent  diurnal 
motion  is  exactly  the  same ;  -iS^rises  in  the  east,  describes 
an  arc  in  the  sky,  sets  in  the  west,  and  reappears  again  in 
the  east  next  morning.  But  there  is  a  difference ;  the  time 
taken  by  him  to  make  a  complete  revolution  is  not  23b  56m  4s, 
but  24  hours,  or  about  4  minutes  longer  than  for  the  fixed 
stars.  This  can  be  verified  roughly  by  waiting  any  day 
until  one  edge  of  the  sun  is  in  a  line  with  a  vertical  wall, 
or,  better  still,  with  two  vertical  strings  hung  with  weights 
at  the  ends  of  them.  The  interval  which  elapses  until  he 
gets  into  the  same  position  next  day  is  then  noted,  and  is 
found  to  be  about  24  hours ;  whereas,  as  we  have  seen  above, 
the  same  experiment  applied  to  a  fixed  star-would  give 
23h  56m  4s. 

The  apparent  position  of  the  sun  among  the  fixed  stars 
cannot  therefore  be  the  same  from  day  to  day,  but  must  be 
slowly  shifting  from  west  to  east,  so  that  he,  as  it  were,  hangs 


6  PROPERTIES  OF  THE  SPHERE.  [CHAP.  I, 

behind  the  fixed  stars  in  his  diurnal  revolution  from  east 
to  west,  taking  a  slightly  longer  time  to  complete  the 
circuit. 

If  we  could  observe  the  fixed  stars  during  the  day-time 
with  our  naked  eyes  as  can  be  done  through  an  astronomical 
telescope,  we  could  actually  see  the  sun's  slow  change  of 
position  from  west  to  east,  and  also  that  he  returns  to  the 
same  position  among  the  fixed  stars  at  the  end  of  a  period  of 
time  which  is  called  a  year.  But  we  can  demonstrate  it 
without  the  aid  of  a  telescope,  thus  : — 

If  we  note  each  evening  some  group  of  stars  which  sets 
in  the  west  some  time  after  sunset,  it  will  be  seen,  if  tha 
observations  be  continued  for  some  weeks,  that  the  interval 
that  elapses  between  sunset  and  the  disappearance  of  the 
stars  becomes  shorter  and  shorter  until  eventually  the  stars 
set  before  the  sun,  and  therefore  cease  to  be  visible  in  the 
early  part  of  the  night.  If,  however,  we  get  up  before  dawn 
and  look  towards  the  eastern  part  of  the  sky  we  will  see  that 
these  stars  have  risen  before  the  sun. 

But  if  we  continue  our  observations  for  365  days,  we 
shall  find  at  the  end  of  that  time  the  sun  in  the  same 
position  relatively  to  the  fixed  stars  as  before,  and  the 
group  in  question  will  again  be  visible  in  the  early  part  of 
the  night.  We  thus  say  that  the  sun  has  two  apparent 
motions : — 

(1)  A  daily  revolution  from  east  to  west  in  common  with 
all  the  heavenly  bodies. 

(2)  A  slow  yearly  revolution  from  west  to  east  among 
the  fixed  stars. 

7.  The  moon. — Besides  its  apparent  diurnal  motion, 
the  moon  also  appears  to  move  from  west  to  east  among 
the  fixed  stars,  but  much  more  quickly  than  the  sun,  as  it 
seems  to  complete  a  revolution  relative  to  the  sun  and  earth 
in  a  period  of  time  which  is  called  a  month. 


CHAP.  I.]  DEFINITIONS.  7 

8.  The    Planets. — Besides  the  fixed    stars,   sun  and 
moon,  there  are  five  other  bodies  visible  to  the  naked  eye 
whose  apparent  motions  among  the   fixed  stars  are,  as  it 
were,  so  whimsical,  that  it  is  difficult  to  reduce  them  to  any 
general  laws.     On  this  account  they  are  called  Planets  or 
wandering  stars.     Sometimes  they  appear  to  move  among 
the  fixed  stars  in  the  same  direction  as  the  sun  and  moon, 
when  their  motion  is  said  to  be  direct :  sometimes  in  the 
opposite  direction,  when   their  motion   is  retrograde,   and 
occasionally  for  a  short  time  they  appear  stationary  among 
the  fixed  stars. 

A  fixed  star  to  the  naked  eye  burns  with  a  twinkling 
light,  a  planet  shines  with  a  steady  light.  Also  when  a  fixed 
star  is  examined  through  even  the  most  powerful  telescope 
it  does  not  seem  increased  in  size,  the  only  difference  being 
that  its  brilliancy  is  intensified.  On  the  other  hand,  the 
disc  of  a  planet  appears  enlarged  when  seen  through  a 
telescope. 

Four  of  the  planets,  Venus,  Mars,  Jupiter,  and  Saturn, 
are  as  bright  or  brighter  than  the  most  brilliant  fixed  star. 
If  the  planet  appear  shining  in  the  south  it  is  Mars,  Jupiter, 
or  Saturn.  Yenus  is  an  evening  or  a  morning  star ;  in 
the  former  case  it  is  seen  in  the  west  after  sunset,  in  the  latter 
case  in  the  east  before  sunrise. 

The  only  other  heavenly  bodies  visible  at  any  time  to  the 
ordinary  observer  are  comets  and  shooting  stars  or  meteors, 
which  complete  the  list  of  those  bodies  of  which  the  follow- 
ing chapters  give  a  more  detailed  description. 

9.  The  Ptolemaic  System. — The   different   celestial 
phenomena  mentioned  above — the  diurnal  revolution  of  all 
heavenly  bodies,  the  yearly  path  of  the  sun,  the  monthly 
motion  of  the  moon,  and  the  apparently  irregular  courses 
taken  by  the  planets  were  accounted  for  by  Ptolemy,  who 
lived  in  the  second  century  after  Christ,  on  apparently  so 


8  PROPERTIES  OF  THE  SPHERE.  [CHAP.  I. 

satisfactory  a  basis  that  it  was  not  until  the  sixteenth 
century  that  the  true  explanation  was  accepted. 

The  whole  celestial  sphere  was  supposed  to  revolve  round 
an  axis  passing  through  the  north  and  south  celestial  poles, 
and  called  the  celestial  axis,  the  earth  being  in  the  centre. 
The  sun,  besides  this  daily  revolution  with  the  celestial 
sphere,  was  supposed  to  have  a  motion  of  its  own  in  the 
opposite  direction  on  the  sphere  describing  a  circle  round  the 
earth  as  centre  once  every  year,  the  moon  similarly  com- 
pleting a  circuit  once  every  month.  The  retrograde  and 
stationary  stages  in  a  planet's  motion  they  accounted  for  in 
a  rather  ingenious  way  by  supposing  the  planets  to  describe 
circles  round  the  sun,  which  in  its  turn  described  a  circle 
round  the  earth. 

The  Copernican  System. — The  true  explanation, 
however,  first  given  by  Copernicus,  and  now  so  well  known, 
shows  that  the  diurnal  revolution  of  the  heavens  is  only 
apparent,  and  that  it  is  really  the  earth  which  rotates  in  the 
opposite  direction,  from  west  to  east,  round  an  axis  which, 
if  produced  out,  would  pass  through  the  north  and  south 
celestial  poles,  thus  causing  the  plane  of  the  observer's 
horizon  by  its  motion  to  uncover  and  bring  into  view  new 
stars  in  the  east,  when  these  stars  are  said  to  rise,  and  to 
cover  up  and  hide  from  view  other  stars  in  the  west  which 
are  then  said  to  set. 

Copernicus  also  referred  the  apparent  yearly  motion  of 
the  sun  round  the  earth  to  a  motion  of  the  earth  round  the 
sun,  and  showed  that  the  earth  and  planets  form  one  system 
revolving  round  tho  sun,  from  which  they  derive  their  light 
and  heat. 

This  explanation  was  at  first  received  by  astronomers 
with  tho  greatest  suspicion,  and  it  was  only  subsequent 
discoveries  which  placed  it  beyond  any  doubt ;  we  should 
therefore  not  take  it  for  granted,  but  examine  carefully  the 
successive  steps  which  led  up  to  these  conclusions.  Although 


CHAP.  I.]  DEFINITIONS.  9 

the  diurnal  revolution  of  the  earth  on  its  axis  was  not 
generally  believed  in  until  about  three  centuries  ago,  it  is, 
however,  not  by  any  means  a  new  idea.  Cicero  mentions 
that  it  was  the  opinion  of  Hicetas  of  Syracuse,  who  lived 
400  years  before  Christ.  Copernicus  says  that  it  was  this 
statement  of  Cicero's  which  first  led  him  to  consider  the 
earth's  motion. 

10.  For  the  purpose  of  reference,  the  relative  positions 
of  the  different  fixed  stars  and  the  apparent  path  of  the  sun 
among  them  are  mapped  out  on  the  surface  of  a  globe,  such 
that  the  arc  joining  the  positions  of  any  two  stars  on  the 
surface  of  this  miniature  globe  subtends  the  same  angle  at 
its  centre  as  the  two  stars  in  question  subtend  at  the  eye 
of  the  observer.  Such  a  celestial  globe,  the  observer  being 
supposed  at  its  centre,  would  then  serve  as  a  representation 
of  the  appearance  of  the  heavens. 

The  reader  should  bear  in  mind  that  such  a  celestial 
globe  only  represents  the  angular  distances  of  the  heavenly 
bodies  from  one  another,  and  not  their  distances  from  the 
earth :  for  the  fixed  stars  are  immensely  further  distant  than 
the  sun  or  planets,  while  being  on  the  surface  of  a  globe 
they  are  represented  as  being  at  the  same  distance  from  the 
observer. 

Definitions. 

(1)  The  great  circle  in  which  the  plane  of  the  horizon 
cuts  the  celestial  sphere  is  called  the  celestial  horizon. 

N.B. — At  sea,  it  is  easy  to  observe  the  position  of  a 
heavenly  body  with  respect  to  the  horizon ;  but  on  land,  on 
account  of  the  inequalities  of  the  earth's  surface,  the  horizon 
cannot  be  seen ;  but  we  can  determine  it  by  taking  a  plane 
perpendicular  to  the  direction  in  which  a  plumb  line  hangs, 
and  note  the  position  of  the  body  with  reference  to  that 
plane.  Tfye  surface  of  a  small  portion  of  liquid  at  rest,  such 
as  mercury,  is  also  used  by  astronomers  to  determine  the 
plane  of  the  horizon. 


10  PROPERTIES  OF  THE  SPHERE.  [CHAP.  I. 

(2)  If  we  imagine  the  direction  of  a  plumb  line  produced 
upwards,  that  point  in  which  it  would  cut  the  celestial  sphere 
is  called  the  zenith. 

(3)  If  we  imagine  the  direction  of  the  plumb  line  pro- 
duced downwards  so  as  to  cut  the  celestial  sphere  in  the 
diametrically  opposite  point,  that  point  is  called  the  nadir. 

N.B. — It  is  evident  that  the  zenith  and  nadir  are  the 
two  poles  of  the  celestial  horizon. 

(4)  Celestial  meridian. — That  great  circle  in  the  heavens 
drawn  through  the  zenith  and  celestial  pole  is  called  the 
meridian. 

(5)  Great  circles  drawn  perpendicular  to  the  horizon,  i.e. 
secondaries  to  the  horizon,  are  called  verticals. 

That  vertical  drawn  due  east  and  west  at  right  angles  to 
the  meridian  is  called  the  prime  vertical. 

(6)  The  four  points  in  which  the  meridian  and  prime 
vertical  cut  the  horizon  are  called  the  four  cardinal  points — 
the  north,  south,  east,  and  west  points. 

(7)  The  celestial  equator  is  that  great  circle  in  the  heavens 
whose  plane  is  at  right  angles  to  the  direction  of  the  celes- 
tial pole. 

The  north  and  south  celestial  poles  are  evidently  the 
poles  of  the  equator. 

The  small  circles  described  round  the  celestial  pole  by 
the  stars  in  their  apparent  diurnal  motion  are  all  parallel  to 
the  celestial  equator. 

It  is  evident  since  any  two  great  circles  bisect  each  other 
(having  a  common  diameter)  that  one-half  of  the  celestial 
•equator  is  above  the  horizon  and  half  below,  so  that  if  any 
star  or  other  heavenly  body  be  situated  in  the  equator  it 
will,  in  its  diurnal  revolution,  remain  equal  times  above  and 
below  the  horizon,  rising  at  the  east  point,  and  setting  at 
the  west  point. 


CHAP.  1.] 


DEFINITIONS. 


11 


FIG.  3. — DIAGRAM  OF  CELESTIAL  SPHERE,  THE  OBSERVER  BEING  AT  ABOUT  THE 
LATITUDE  OF  DUBLIN. 

ASCB  is  the  diurnal  parallel  of  a  star  S,  A  being  the  point  where  the  star 
rises,  B  where  it  sets,  and  C  the  point  where  it  crosses  the  meridian. 


0    . 

.     Observer. 

R    .     . 

North  Point 

HmEn    . 

.     Horizon. 

jff    .     . 

South  Point. 

Z    . 

.     Zenith. 

Wl 

East  Point. 

P    . 

.     Qelestial  Pole. 

n     .     . 

West  Point. 

HZPR     . 

.     Meridian. 

8    .     . 

A  Star. 

£mQn     . 

.     Celestial  Equator. 

ZSK    .     . 

Vertical  through  S. 

mZn    . 

.     Prime  Vertical. 

LSPZ    .     . 

Hour  angle  of  S. 

(8)  Definition. — The  ecliptic  is  the  apparent  path,  of  the 
sun  among  the  fixed  stars  in  the  course  of  a  year. 

When  this  apparent  annual  path  of  the  sun  is  traced  out 
on  the  celestial  sphere  it  is  found  that  it  can  be  represented 
by  a  great  circle.  Its  name  arises  from  the  fact  that  if  the 
moon  in  her  monthly  revolution  happen  to  cross  the  plane 
of  the  ecliptic  when  it  is  full  or  new  moon,  there  will  be  an 
eclipse,  in  the  former  case  of  the  moon,  and  in  the  latter  of 
the  sun. 

Obliquity  of  Ecliptic  to  Equator.     Equinoxes. 

11.  The  angle  at  which  the  planes  of  the  ecliptic  and 
equator  cut  is  about  23°  28'  which  is  called  the  obliquity  of 
the  ecliptic  to  the  equator.  These  two  great  circles  must 


12  PROPERTIES   OF   THE    SPHERE.  [CHAP.  I. 

intersect  in  two  points,  therefore  on  two  days  each  year  the 
sun  is  in  the  act  of  crossing  the  equator,  and  on  those  days 
his  diurnal  path  almost  coincides  with  the  equator,  rising  due 
east  and  setting  due  west  (fig.  3);  one-half  of  his  diurnal  path 
is  therefore  above  horizon  and  half  below,  and  day  and  night 
are  of  equal  duration  all  over  the  world.  From  this  latter 
circumstance  these  two  periods  are  called  the  Equinoxes,  the 
two  points  of  intersection  of  the  ecliptic  with  the  equator 
being  called  the  equinoctial  points.  One  of  these  points  is  called 
the  first  point  of  Aries  (T),  the  other  the  first  point  of  Libra 
(,£±)9  because  when  these  points  were  first  named  by  the 
ancient  astronomers  they  were  in  the  constellations  of  Aries 
and  Libra  respectively.  The  sun  is  at  the  first  point  of  Aries 
on  the  21st  March  when  crossing  from  the  south  to  the  north 
side  of  the  equator  ;  this  date  is  called  the  vernal  or  spring 
equinox,  and  is  at  Libra  on  the  23rd  September,  in  his  passage 
from  the  north  to  the  south  side  of  the  equator,  this  date 
being  the  autumnal  Equinox. 

The  Signs  of  the  Zodiac. 

12.  Ancient  astronomers  found  by  observation  that  the 
moon  and  planets  were  never  at  any  time  at  a  very  great 
angular  distance  from  the  ecliptic;  they  therefore  conceived  an 
imaginary  belt  in  the  heavens  extending  for  about  8°  on  either 
side  of  the  ecliptic.  Inside  this  space  the  moon  and  planets, 
and  of  course  the  sun  were  always  to  be  found.  They  called 
this  belt  the  zodiac  from  their  imagining  certain  forms  of 
animals  situated  within  it,  which  they  named  the  signs  of  the 
zodiac.  There  are  twelve  signs  of  the  zodiac:  these  together 
with  the  symbols  which  represent  them  are  as  follows  : — 

Aries.    Taurus.      Gemini.  Cancer.  Leo.        Virgo. 

T  8  n  ©  Si  WSL 

Libra.    Scorpio.    Sagittarius.    Capricornus.     Aquarius.     Pisces. 
-£=         m  £  V?  zz  K 


CHAP.  I.]  DEFINITIONS.  13 

Altitude,  Azimuth. 

13.  The  altitude  of  a  heavenly  body  is  its  distance  from 
the  horizon  measured  on  the  arc  perpendicular  to  the  horizon 
drawn  through  the  body  (i.  e.  on  the  vertical  drawn  through 
the  body). 

'  The  azimuth  of  a  body  is  the  arc  intercepted  on  the 
horizon  between  the  foot  of  the  vertical  drawn  through 
the  body  and  the  meridian. 

Thus  (fig.  3)  SK=  altitude,  and  HK  =  azimuth  of  the 
star  S. 

Of  course  it  is  immaterial  whether  we  call  RK  or  HK 
the  azimuth  of  the  star  S,  provided  we  mention  whether 
we  are  measuring  it  from  the  north  or  south  point.  In 
northern  latitudes  the  azimuth  is  generally  measured  east 
and  west  from  the  south  point,  and  in  southern  latitudes 
from  the  north  point.  Thus  if  the  arc  HK=  30°,  the  azimuth 
of  8  =  30°  E. 

The  arc  SZis  called  the  zenith  distance  of.  the  body,  and  is 
evidently  the  complement  of  the  altitude. 

The  position  of  a  body  on  the  celestial  sphere  with  respect 
to  the  observer's  horizon  and  meridian  can  be  described  by 
knowing  its  altitude  and  azimuth,  but  as  the  horizon  of  the 
observer  is,  owing  to  the  earth's  rotation,  changing  every 
instant,  and,  moreover,  both  the  horizon  and  meridian  are 
different  for  different  places  on  the  earth,  therefore  the 
altitude  and  azimuth  of  a  heavenly  body  only  describe  its 
position  at  some  particular  instant  and  observed  from  a 
certain  definite  place  on  the  earth. 

Declination  and  Right  Ascension. 

14.  Instead  of  describing  the  position  of  a  body  with 
reference  to  the  horizon  we  may  refer  it  to  the  equator.  The 
measurements  by  which  its  position  is  then  indicated  are 


14  PROPERTIES  OF  THE  SPHERE.  [CHAP'  *• 

independent  of  the  position  of  the  observer  on  the  earth,  and 
do  not  change  appreciably  each  instant,  but,  as  we  shall 
subsequently  see,  only  after  comparatively  long  periods  of 
time. 

The  declination  of  a  heavenly  body  is  its  distance  from  the 
equator  measured  on  an  arc  perpendicular  to  the  equator 
drawn  through  the  body. 

The  right  ascension  is  the  arc  of  the  equator  intercepted 
between  the  firsLpwntfef  Aries  and  the  perpendicular  to 
the  equator  drawn  through  the  body. 

The  right  ascension  is  reckoned  from  T  eastward  from 
0°  to  360°. 

Thus  (fig.  4)  let  JSQ  re- 
present the  equator,  AB  the 
ecliptic;  draw  a  common 
secondary  AQBP  to  both 
passing  through  P  the  pole 
of  the  equator  (celestial 
pole)  and  Pf  the  pole  of 
the  ecliptic.  Then  if  S  be 
the  position  of  a  heavenly 
body  we  have : 

FIG.  4. 

Sm  =  declination  of  body  measured  along  PM. 

rm  =  right  ascension  of  body  measured  along  the  equator. 

The  arc  SP  is  called  the  polar  distance  of  the  body  and 
is  evidently  the  complement  of  the  declination. 

Celestial  Latitude  and  Longitude. 

The  position  of  a  body  may  be  indicated  also  with 
reference  to  the  ecliptic. 

The  latitude  of  a  heavenly  body  is  its  distance  from  the 
ecliptic  measured  on  a  perpendicular  arc  to  the  ecliptic. 

The  longitude  is  the  arc  of  the  ecliptic  intercepted  between 


CHAP.  I.]  DEFINITIONS.  15 

the  first  point  of  Aries  and  a  perpendicular  arc  to  the  ecliptic 
drawn  through  the  body. 

Thus  (fig.  4)  Sn  =  latitude  of  S  and  T»  =  longitude. 

The  terms  celestial  latitude  and  longitude  are  applied  to 
these  measurements  to  distinguish  them  from  terrestrial 
latitude  and  longitude  with  which  they  are  not  in  any 
way  connected. 

The  longitudes  of  heavenly  bodies  are,  like  their  right 
ascensions,  measured  from  T  eastward  from  0°  to  360°. 

Both  the  declinations  and  latitudes  of  heavenly  bodies 
vary  from  0°  to  90°  on  either  side  of  the  equator  and  ecliptic 
respectively.  They  are  counted  north  or  south  according  to 
whichever  celestial  pole  happens  to  be  on  the  same  side  of  the 

great  circle  from  which  they  are  measured. 

\ 

Decimation  Circles.    Hour  Angle.         ^ 

Secondaries  to  the  equator  are  called  declination  circles 
because  it  is  on  these  circles  that  the  declinations  of  heavenly 
bodies  are  measured. 

The  angle  which  the  declination  circle  through  a  star 
makes  with  the  meridian  is  called  the  hour  angle  of  the  star, 
because  when  this  angle  is  known  we  are  able  to  calculate  the 
time  that  must  elapse  before  the  star  crosses  the  meridian  or 
the  time  which  has  elapsed  since  it  last  crossed  it,  from  the 
fact  that  the  star  completes  a  revolution  of  360°  round  the 
celestial  pole  in  23*  56m  48. 

Thus  (fig.  3)  the  angle  SPZ  =  hour  angle  of  star  8. 

Declination  circles  are  on  this  account  also  called  hour 
circles. 

Changes  in  the  Sun's  Declination  during  the  year  as  he  describes 
the  Ecliptic. 

15.  At  the  spring  equinox  the  declination  of  the  sun  is 
zero,  he  being  at  v  (fig.  4).  Each  day,  however,  on  account 


16  PROPERTIES  OF  THE  SPHERE.  [CHAP.  I. 

of  his  slow  annual  motion  his  declination  increases  until, 
some  time  about  21st  June,  he  reaches  his  greatest  declina- 
tion, viz.  the  arc  BQ,  (fig.  4).  But  BQ  is  the  arc  intercepted 
by  the  ecliptic  and  equator  on  their  common  secondary,  and 
must  therefore  measure  the  angle  between  those  great  circles, 
which  is  23°  28'. 

Therefore  the  arc  BQ  =  greatest  declination  of  sun  at  mid- 
summer =  23°  28'  north.  This  period  is  called  the  summer 
solstice  (sol,  stare],  because  the  sun  before  descending  to  •£= 
seems  for  some  time  to  stand  still. 

After  midsummer  the  sun's  declination  gradually  de- 
creases until  at  ===  (about  23rd  September)  it  is  again  zero. 
Between  23rd  September  and  the  21st  of  the  following 
March  the  declination  of  the  sun  is  south,  reaching  at 
mid-winter  (21st  December)  a  value  AE  which  =  23°  28' 
south.  This  period  is  called  the  winter  solstice;  therefore 
we  have : — 

The  sun's  declination  at  the  vernal  equinox    =    0 

summer  solstice     =  23°28'N, 
„  „  autumnal  equinox  =     0 

„  „  winter  solstice        =23°28'S. 

During  this  period  which  is  called  a  year  the  sun's  right 
ascension  and  longitude  increase  from  0°  at  vernal  equinox 
to  360°  immediately  before  the  following  vernal  equinox, 
each  being  90°  on  21st  June,  180°  on  23rd  September,  and 
270°  on  21st  December. 

It  is  hardly  necessary  to  mention  that  the  sun  being 
in  the  ecliptic  has  his  latitude  always  zero. 

N.B. — When  we  speak  of  the  sun's  declination,  &c.,  we 
mean  that  of  the  centre  of  the  sun's  disc. 

Tropics  of  Cancer  and  Capricorn. 

If  on  the  celestial  sphere  we  draw  two  small  circles 
parallel  to  the  equator,  and  distant  from  it  23°  28'  north 


I.]  DEFINITIONS.  17 

and  south,  these  small  circles  will  nearly  coincide  with 
the  sun's  apparent  diurnal  path  on  21st  June  and  21st 
December.  They  are  called  the  tropics  because  the  sun 
seems  to  be  on  the  point  of  turning  at  these  periods.  The 
northern  circle  is  the  Tropic  of  Cancer,  the  southern  the 
Tropic  of  Capricorn. 

The  Equinoctial  Colure  is  the  secondary  to  the  equator 
passing  through  the  equinoctial  points. 

The  Solstitial  fjolure  is  the  secondary  to  the  equator 
passing  through  the  solstices,  and  hence  is  also  a  secondary 
to  the  ecliptic. 

The  altitude  of  a  Star  is  greatest  when  on  the  Meridian. 

Let  S  represent  the  star  in  the  meridian,  and  8'  its  position 
at  any  other  time.  Join  ZS'  and  PS'.  Now  since  any  two 


sides  of  a  spherical  triangle  are  together  greater  than  the 
third,  therefore  we  have  j 


but  PS'  =  PS,  since  a  star  always  maintains  the  same  dis- 
tance from  the  pole  ; 

.-.  ZS'  +  ZP>PS. 

Take  away  the  common  part  ZP,  therefore  we  get  Z& 

c 


18  PROPERTIES  OF  THE  SPHERE.  [cHAP.  I. 

greater  than  ZS,  that  is,  the  zenith  distance  is  least  when  on 
the  meridian,  and  hence  the  meridian  altitude  is  greatest. 

In  the  same  way  it  can  be  shown  that  the  depression  of  a 
body  below  the  horizon  is  greatest  when  on  the  meridian. 

EXERCISES. 

\ 

1.  What  are  the  altitude  and  hour  angle  of  the  zenith  ?  Ans.  90° :  0. 

2.  What  are  the  declination  and  latitude  of  the  celestial  pole  ? 

Ans.  90:  66°  32' (90 -23°  28') 

3.  How  far  is  the  pole  of  the  ecliptic  from  the  celestial  pole,  or,  in  other 
words,  what  is  the  magnitude  of  the  arc  PF  in  fig.  4  ?  Ans.  23°  28'. 

4.  What  are  the  declination,  right  ascension,  latitude,  and  longitude  of  ^  ? 

Ans.  0  :  180° :  0 :  180°. 

5.  What  point  in  the  heavens  has  its  declination,  right  ascension,  latitude, 
nd  longitude  each  equal  to  zero  ?  Ans.  First  point  of  Aries  ( T ) . 

6.  If  a  certain  star  cross  the  meredian  at  11  o'clock  P.M.  to-night,  at  what 
o'clock  will  it  cross  the  meridian — (1)  to-morrow  night;    (2)  15  days  hence, 
assuming  the  sun's  change  of  right  ascension  throughout  the  year  to  be  uniform  ? 
See  Arts.  (5)  and  (6).)  Ans,  (I)  About  10.56  P.M. 

(2)    About  10  P.M. 

7.  At  what  hour  will  the  same  star  cross  the  meridian  a  year  hence  ? 

Ans.  11  P.M.  again. 

8.  A  star  is  in  the  meridian  10°  above  the   pole  at   midnight  to-night, 
where  will  it  be  at  midnight — (1)  six  months  hence  ;  (2)  a  year  hence,  sup- 
posing the  sun's  apparent  motion  in  the  ecliptic  to  be  uniform  ? 

9.  What  is  the  sun's  right  ascension  on  21st  March,  21st  'June,  23rd  Sep- 
ember,  21st  December.  Ans.  0  :  90° :  180°  :  270°. 

10.  Calculate  what  would  be  the  declination  and  right  ascension  of  the  sun 
on  21st  April  if  the  changes  in  these  quantities  were  uniform  throughout  th 
year.  Ans.  7°  49'  20"  N. :  30°. 

11.  Making  the  same  assumption  as  in  the  last  question— (1)  Find  at  what 
ime  the  sun's  right  ascension  should  be  120° ;  (2)  at  what  time  should  his 
eclination  be  15°  38'  40"  N.  Ans.  (I)  21st  July. 

(2)  21st  May  or  21st  July. 

N.B. — The  reader  can,  by  reference  to  a  celestial  globe  or  the  Nautical 
Almanac,  see  that  the  results  obtained  in  Examples  (10)  and  (11)  are  not  the 
correct  values  of  the  right  ascension  and  declination  of  the  sun  on  the  dates 
mentioned,  which  shows  that  the  changes  in  these  quantities  throughout  the 
year  are  not  at  all  uniform. 

12.  What  is  the  time  of  .sunrise  and  sunset  at  any  place  during  the  equi- 
noxes? A_ns.  About  6  A.M.  and  6  P.M. 

13.  What  is  the  hour  angle  of  the  sun  at  sunrise  on  21st  March  ?     Ans.  90°. 


CHAPTER  II. 

THE   EARTH. 

16.  THAT  the  earth's  shape  is  approximately  spherical 
lias  been  known  from  the  earliest  times.     It  will  not  here 
be  necessary  to  do  more  than  mention  the  different  reasons 
which  lead  us  to  this  conclusion.     They  are : — 

(1)  The  hull  of  a  ship  disappears  first,  which  shows  that 
the  ship  is  sailing  on  a  convex  surface. 

(2)  The  outline  of  the  earth's  shadow,  as  seen  on  the 
surface  of  the  moon  during  an  eclipse,  always  seems  an  arc 
of  a  circle,  and  no  body  but  a  sphere  can  project  a  circular 
shadow  in  all  positions. 

(3)  The  most  conclusive  proof,  however,  depends  on  the 
fact,  which  is  found  by  observation,  that  equal  distances  gone 
over  by  the  observer  due  north    or  south  produce  almost 
equal  variations  in  the  meridian  altitude  of  any  chosen  star 
(or  of  the  celestial  pole).      This  could  not  happen  except  on 
the  supposition  that  the  earth  is  nearly  spherical. 

Celestial  Pole  Constant  in  Direction. 

17.  The  celestial  pole  being  supposed  to  be  situated  at  an 
indefinitely  great  distance  away  comp  ared  with  any  distance 
on  the  earth,  therefore,  as  the  observer  changes  his  position 
on  the  earth's  surface,  the  lines   drawn  from  those  posi- 
tions in  the  direction  of  the  celestial   pole  are  practically 
parallel. 

C2 


20  THE   EARTH.  [CHAP.  II. 

Earth's  Axis.     Terrestrial  Equator.     Terrestrial  Latitude  and 

Longitude. 

That  diameter  of  the  earth  which  is  parallel  to  the 
constant  direction  of  the  celestial  pole  is  called  the  earth's 
axis. 

The  earth's  axis  cuts  the  surface  of  the  earth  in  two 
points  called  the  north  and  south  poles  of  the  earth. 

That  great  circle  drawn  round  the  earth  whose  plane  is 
perpendicular  to  the  earth's  axis  is  called  the  terrestrial 
equator. 

Great  circles  drawn  through  the  poles  of  the  earth  are 
called  terrestrial  meridians. 

Therefore,  every  place  on  the  earth's  surface  may  be 
supposed  to  have  its  meridian. 

The  meridian  of  Greenwich  is  called  the  first  meridian. 

The  latitude  of  a  place  is  its  distance  north  or  south 
of  the  equator  measured  on  the  meridian  through  the 
place. 

The  longitude  of  a  place  is  its  distance  east  or  west  of  the 
first  meridian,  and  is  measured  by  the  number  of  degrees  in 
the  arc  intercepted  on  the  equator  between  the  meridian  of 
the  place  and  the  first  meridian. 

All  places  situated  on  the  same  parallel  to  the  equator 
have  evidently  the  same  latitude,  and  situated  on  the  same 
meridian  have  the  same  longitude.  Latitude  is  measured 
north  and  south  from  0°  to  90°,  and  longitude  east  and  west 
from  0°  to  180°. 

Corresponding  to  the  Tropics  of  Cancer  and  Capricorn 
on  the  celestial  sphere,  we  imagine  two  small  circles  on  the 
earth  parallel  to  the  equator,  one  north  the  other  south,  and 
distant  from  it  about  23°  28' :  these  small  circles  are  also 
called  the  Tropics  of  Cancer  and  Capricorn.  The  two  small 
circles  drawn  round  the  north  and  south  poles  of  the  earth  at 


DIRECTION  OF  POLE. 


CHAP.  ,11.]  ALTITUDE  OF  CELESTIAL  POLE.  21 

a  distance  of  23°  28'  are  called  the  arctic  and  antarctic  circles 
respectively. 

The  portion  of  the  earth's  surface  enclosed  between  the 
two  tropics  is  called  the  torrid  zone,  between  the  tropics 
and  the  arctic  and  antarctic  circles  the  temperate  zones,  and 
between  the  arctic  and  antarctic  circles  and  the  poles  the  frig  id 
zones. 

18.  The  altitude  of  the  celestial  pole  at  any  place  is  equal  to 
the  latitude  of  the  place. 

For  let  0  be  the  position  of  the  observer ;  EOQ,  the  meri- 
dian of  the  place,  cutting  the  equator  in  E  and  Q.  If  OP 
represent  the  direction  of  the 
celestial  pole  as  seen  from  0> 
then  the  line  CP  drawn  from 
C,  the  centre  of  the  earth,  in 
the  direction  of  the  celestial 
pole,  will  be  parallel  to  OP 
(the  pole  being  so  far  dis- 
tant). The  horizon  of  the 
observer  will  be  represented 
by  a  tangent  plane  OH  drawn 
to  the  earth  at  0. 

Then  we  have  to  prove 
that  the  angle  9  which  is  the 
altitude  of  the  pole  =  the  arc 
EO  or  the  angle  0,  which  is  FlG-  6< 

the  latitude  of  the  place.  Since  OP  is  parallel  to  CP,  the 
angle  a  =  the  angle  j3 ;  but  0  is  the  complement  of  a,  and  0  is 
the  complement  of  j3 ;  therefore  9  =  0,  or  altitude  of  pole 
=  latitude  of  place.  From  this  it  follows  that  the  change  in 
the  altitude  of  the  pole  must  equal  the  change  in  the  lati- 
tude of  the  observer  as  he  proceeds  north  or  south. 


22  THE  EARTH.  [CHAP.  II. 

Length  of  a  degree  of  Latitude.    Magnitude  of  Earth. 
Shape  of  Earth. 

19.  The  measurement  of  the  length  of  a  degree  of  lati- 
tude on  the  earth  is  an  operation  of  much  practical  difficulty. 
A  position  is  chosen  on  the  earth,  and  the  altitude  of  the 
pole  observed.  Another  station  is  chosen  due  north  or  south 
of  the  former  position  at  such  a  distance  from  it  that  the 
altitude  of  the  pole  is  increased  or  diminished  by  1°,  as  the 
case  may  be.  The  length  of  the  arc  of  the  meridian  between 
the  two  stations  is  then  measured,  and  is  found  to  have  a 
mean  value  of  about  69TV  miles,  which  must  be  the  length  of 
a  degree.  The  length  of  a  degree  has  thus  been  calculated 
at  about  twenty  different  places  on  the  earth,  and  the  results 
have  not  been  found  to  differ  to  any  very  great  extent, 
which  is  confirmatory  evidence  of  the  earth's  approximate 
spherical  shape. 

It  has,  however,  been  found  that  the  length  of  a  degree 
near  the  poles  is  somewhat  greater  than  near  the  equator, 
which  shows  that  the  curvature  of  the  earth  is  not  so  great 
at  the  poles  as  at  the  equator,  or,  in  other  words,  that  the 
earth  is  slightly  flattened  at  the  poles.  In  fact  the  figure  of 
the  earth  is  what  is  called  an  oblate  spheroid,  differing  but 
little  from  a  sphere.  The  lengths  of  a  degree  of  latitude 
at  different  parts  of  the  earth  have  been  found  to  be  as 
fol  ows : — 

At  the  equator,  .  .  68704  miles. 

At  latitude  20°,  .  .  68-786  „ 

„  „  40%  .  .  68-993  „ 

„  „  60°,  ,  .  69-230  „ 

„        „      80°,    .     .     69-386      „ 

i 

The  length  of   a   degree  being   about  69TV  miles,  art 


CHAP.  II.] 


MAGNITUDE  OF  EARTH. 


approximate  value  for  the   circumference  and  diameter  of 
the  earth  can  thus  be  found.* 

1°  =  69TV  miles  ; 

/.  360°  =  somewhat  under  25,000  miles  =  circum- 
ference of  earth. 

Diameter  of  earth  =         ^  =  somewhat  under  8000  miles. 

The  polar  diameter  of  the  earth  is  found  to  be  about  26  miles 
shorter  than  the  equatorial. 

Appearances  of  the  Celestial  Sphere  due  to  Observer's  Change 

of  Place  on  Earth. 

20.  If  the  observer,  starting  from  any  place  north  of  the 
equator,  move  due  north  along  the  meridian  of  the  place, 
the  celestial  pole  will  appear  to  rise  in  the  sky  as  his  latitude 
increases  (Art.  18).  If  he  reach  the  north  pole  the  celestial 
pole  will  appear  right  overhead  in  the  zenith ;  the  celestial 
equator  will  therefore  coincide  with  his  horizon  (see  fig.  7). 
The  apparent  diurnal  paths  of  the  stars  will  appear  as  small 
circles  parallel  to  the  horizon ; 
therefore,  all  the  stars  visible 
will  be  circumpolar.  Those 
stars,  on  the  other  hand,  whose 
positions  in  the  heavens  are 
south  of  the  celestial  equator 
will  never  rise  into  view. 
Therefore,  an  observer  at  the 
north  pole  will  never  see  more 
than  half  the  heavens,  no  part 
of  which,  however,  ever  sinks 
below  his  horizon.  Such  a 
celestial  sphere  is  called 
parallel  sphere. 


Z-&. 


FIG.    7. — Parallel   Sphere,   observer 
being  at  either  pole  of  Earth. 


is    called    a 
The  sun  for  half  the  year  (21st  March  to 


*  The  above  method  was  that  employed  by  Eratosthenes  (230  B.C.)  in  de- 
termining the  magnitude  of  the  earth,  except  that  he  measured  the  meridian 
altitude  of  the  sun  instead  of  the  altitude  of  the  pole. 


24 


THE  EARTH. 


[CHAP.  II. 


23rd  September)  being  north  of  the  equator,  will  during  this 
period  appear  above  the  observer's  horizon.  For  these  six 
months  he  will  appear  to  make  a  circuit  every  24  hours  in  the 
heavens,  which  would  be  parallel  to  the  horizon  but  for  his 
continual  change  of  declination.  His  greatest  altitude  is 
reached  on  the  21st  June,  and  is  then  23°  28'.  For  the 
remaining  six  months  the  sun  keeps  below  the  horizon, 
reaching  a  distance  below  it  of  23°  28'  on  21st  December. 
Therefore,  at  the  north  pole  the  day  and  night  are  each  six 
months  long.  However,  of  the  six  months'  night  at  the 
north  pole  a  considerable  portion  is  twilight. 

Observer  at  Equator. 

Let  the  observer  now  proceed  southwards,  and  he  will 
find  that  the  celestial  pole  gradually  falls  in  the  sky  until  at 
the  equator  it  will  appear  on  the  horizon  coinciding  with  the 
north  point,  the  south  celestial  pole  coinciding  with  the  south 
point.  The  celestial  equator  will  therefore  pass  through  the 
zenith  and  nadir,  and,  cutting  the  horizon  at  right  angles, 
will  coincide  with  the  prime  vertical  (fig.  8). 

The  apparent  diurnal 
paths  of  the  stars  being 
parallel  to  the  celestial 
equator  will  be  bisected  at 
right  angles  by  the  horizon ; 
the  stars  will  therefore  be 
an  equal  time  above  and 
below  the  horizon.  There 
are  therefore  no  stars  cir- 
cumpolar,  but  every  star  in 
the  heavens  appears  for 
nearly  twelve  hours  above 
the  horizon. 


FIG.  8. — Right  Sphere,  observer  being  at 
Equator. 


It  is  evident,  also,  that  day  and  night  are  equal  through- 
out the  year  at  the  equator. 


CHAP.  II.]  PROOFS   OF   EARTH'S   ROTATION.  25 

Since  the  horizon  bisects  the  diurnal  paths  of  the  stars  at 
right  angles,  this  sphere  is  called  a  right  sphere. 

Similarly,  in  the  southern  hemisphere  the  south  celestial 
pole  will  increase  its  altitude  as  the  south  latitude  increases. 


Observer  at  about  Latitude  of  Dublin. 

As  the  latitude  of  Dublin  is  about  53°  20'N.,  the  celestial 
pole  will  have  an  altitude  PR  =  53°  20'  (fig.  3).  The 
apparent  diurnal  paths  of  the  stars  cut  the  horizon  obliquely. 
Some  stars  are  circumpolar,  and  some  rise  and  set,  while  other 
stars  whose  decimations  are  south  become  visible  for  a  small 
portion  of  their  daily  circuit.  The  sun's  apparent  diurnal 
path  during  the  summer  months  may  be  represented  by  the 
circle  A  CB  north  of  the  equator,  of  which  there  is  more  than 
half  above  the  horizon ;  therefore  during  the  summer  we  have 
a  long  period  of  daylight  and  a  short  night.  Also  in  the  winter, 
when  the  sun's  declination  is  south,  we  have  a  short  period  oj 
daylight  and  a  long  night. 

The  sphere  in  this  position  is  called  an  oblique  sphere. 

Diurnal  Rotation  of  the  Earth. 

21.  That  the  apparent  diurnal  motion  of  the  heavens 
from  east  to  west  is  really  due  to  the  earth  rotating  round 
an  axis  from  west  to  east  appears  from  the  following  con- 
siderations : — 

(1)  From  simplicity. 

(2)  From  analogy. 

(3)  From  centrifugal  force. 

(4)  The  experiment  of  letting  a  body  fall  from  the  top  of 

a  high  tower. 

(5)  Foucault's  pendulum  experiment. 

Under  the  first  three  of  these  headings  are  comprised  the 
arguments  which  show  that  it  is  extremely  probable  that  the 


26  THE   EARTH.  |_CHAP-n« 

earth  does  rotate  ;  but  (4)  and  (5)  are  experimental  proofs  of 
its  rotation. 

From  Simplicity.  —  At  the  time  of  Copernicus,  the  only 
argument  in  favour  of  the  earth's  rotation  was,  that  this  was 
a  much  simpler,  and  therefore  a  much  more  probable,  expla- 
nation than  that  all  the  stars  and  other  heavenly  bodies 
should  be  connected  in  such  a  complicated  manner  as  to 
perform  each  its  revolution  round  the  celestial  pole  in  the 
same  time. 

From  Analogy.  —  However,  the  subsequent  invention  of 
telescopes  (1609)  supplied  an  additional  argument.  By  the 
aid  of  the  telescope  we  can  see  that  many  of  the  planets,  as 
well  as  the  sun  and  moon,  are  spherical  bodies  rotating  about 
axes,  from  which  we  conclude  that  it  is  probable  the  earth 
also  rotates. 

From  Centrifugal  Force.  —  If  the  sun  and  planets, 
not  to  speak  of  the  fixed  stars,  really  described  circles  of 
such  large  radius  in  such  a  short  period  as  one  day,  it 
would  need  an  enormous  attracting  force  acting  towards 
the  centres  of  those  circles  to  keep  them  from  flying  off 
in  a  tangent. 

For  we  know  from  mechanics  that  if  m  be  the  mass  of  a 
body  moving  in  a  circle  of  radius  r  with  a  periodic  time  =  T, 
the  necessary  force  acting  towards  the  centre  to  keep  it  in  its 
circular  path  would  be  given  by  the  formula  :  — 


But  here  r  would  be  very  large  and  T  very  small,  there- 
fore F  would  be  enormously  great.  But  there  are  no  bodies 
we  know  whose  attractions  could  be  as  great  as  this  ;  therefore 
the  idea  of  the  diurnal  motion  of  the  sun  and  planets,  as  well 
as  the  stars,  is  very  improbable. 


CHAP.  II.] 


. 


\ 

VERSITYl 


of  J 

r>*^ 
PROOFS  or  EARTH'S  ROTATION. 


27 


Q 


Experimental  Proof  from  Falling  Bodies. 

22.  Newton  first  suggested  that  if  the  earth  rotate  from 
west  to  east,  a  body,  on  being  let  fall  from  a  considerable 
height  above  the  earth's  surface,  should  fall  to  the  east  of  the 
vertical  line. 

For,  let  P  be  the  place  from  which  the  body  is  let  drop, 
suppose  the  top  of  a  tower  (fig.  9) ;  PAC  the  vertical 
through  P  passing  through  (7,  the  centre  of  the  earth.  Then, 
if  PQ  represent  the  arc  de- 
scribed by  the  top  of  the  tower 
while  the  body  is  falling,  AB 
will  represent  the  arc  described 
by  the  base  of  the  tower  which, 
being  less  than  PQ,  shows  that 
the  base  moves  with  a  less  velo- 
city than  the  top.  But  while 
the  body  is  falling  in  the  air 
it  preserves  the  same  velocity 
towards  the  east  which  it  had  at 
starting  in  common  with  the 
top  of  the  tower,  which,  being 
slightly  greater  than  that  of 
the  base  of  the  tower,  will  cause  the  body  to  deviate  slightly 
to  the  east.  Thus,  if  we  cut  off  AB'  =  PQ,  B  will  represent 
the  position  of  the  base  of  the  tower,  and  B'  of  the  body, 
when  it  reaches  the  ground. 

If  now  we  let  a  body  fall  from  a  high  tower,  and  we 
find  by  actual  measurement  that  daring  its  fall  it  has  de- 
viated to  the  east  of  the  vertical,  the  only  cause  we  can 
give  for  this  deviation  is  that  the  earth  rotates  from  west 
to  east. 

It  is,  however,  very  difficult  to  perform  the  experiment 
so  as  to  give  a  decided  result,  the  height  of  the  tower 
being  so  small  compared  with  the  radius  of  the  earth  that 


FIG.  9. 


28 


THE   EARTH. 


[CHAP.  II. 


the  deviation  would  be  very  slight.  It  has  been  tried  at 
Boulogne  and  Hamburg,  and  the  deviation  was  found  to 
be  one-third  of  an  inch  in  a  fall  of  250  feet. 


Pendulum  Experiments. 

23.  The  experimental  proof  of  the  earth's  rotation 
which  is  most  striking  is  that  first  performed  by  Foucault 
in  Paris  in  1851,  and  very  many  times  since  by  different 
observers. 

Before  entering  upon  the  details  of  the  experiment,  we 
will  first  suppose  that  the  earth  does  rotate  from  west  to  east, 
and  see  what  effect  this  rotation  would  have  on  a  pendulum 
swinging  at  the  north  pole.  We  kn  ow,  from  mechanics,  that 
if  a  pendulum  vibrate  under  the  action  of  gravity  alone,  the 
plane  of  oscillation  will  remain  fixed  in  space,  for  there  is  no 
force  to  make  it  deviate  from  that  plane. 

Therefore,  if  it  were  possible 
to  have  a  pendulum  vibrating 
at  the  north  pole,  the  observer 
and  the  plane  in  which  he  stands 
would  be  carried  by  the  rota- 
tion of  the  earth  round  the 
fixed  plane  of  the  pendulum 
through  360°  in  23h  56m  4s.  The 
observer,  however,  would  be 
altogether  unconscious  of  his 
own  motion  and  that  of  the 
plane  in  which  he  stands,  and 
it  would  appear  to  him  as  if 
the  plane  of  the  pendulum  turned  round  in  the  opposite 
direction,  making  a  complete  circuit  in  23h  56m  4s  (fig.  10). 

On  the  contrary,  if  a  pendulum  be  set  in  motion  at  the 
equator,  the  plane  of  vibration,  together  with  the  observer 
and  the  surrounding  surface  of  the  earth  will  be  carried 


1 

Equator 

• 

\ 

}  I 

FIG.  10. 


CHAP.  II.] 


PENDULUM  EXPERIMENTS. 


29 


FIG.  II. 


bodily  round  in  one  common  motion;  therefore  there  will 
be  no  disturbance  in  the  re- 
lative positions  of  the  plane 
of    the    pendulum    and    the 
landmarks  about  it  (fig.  11). 

At  a  place  intermediate  be- 
tween the  equator  and  pole, 
the  parts  of  the  earth  in  the 
immediate  neighbourhood  of 
the  pendulum  which  are  near- 
est the  equator  will  have  a 
greater  velocity  towards  the 
east  than  the  parts  nearest  the  pole ;  therefore  the  plane 
in  which  the  observer  stands  will  really  revolve  beneath  the 
pendulum,  or,  in  other  words,  the  plane  of  vibration  of  the 
pendulum  will  seem  to  revolve  in  the  opposite  direction  with 
respect  to  the  observer  and  surrounding  landmarks.  The 
time  of  the  apparent  revolution  of  the  pendulum  will  get 
greater  the  nearer  we  approach  the  equator,  until  on  the 
equator  itself,  as  we  have  seen  above,  the  plane  of  vibration 
does  not  seem  to  change  at  all. 

It  is  easy  to  prove, 
supposing  that  the  earth 
does  rotate,  that  at  a  place 
whose  north  or  south  lati- 
tude is  X,  the  time  of 
apparent  revolution  of 
the  pendulum  would  be  T 
cosec  X,  where  T  =  time  of 
revolution  of  the  earth  on 
its  axis. 

For,  let  Om  be  the  direc- 
tion of  the  observer,  P  the 

north  or  south  pole  of  the  earth,  EQ  the   equator,    and 
X  =  latitude  of  observer. 


30  THE  EARTH.  [CHAP.  II. 

Now,  the  earth  revolves  round  OP  through  360°  in  T 

SfiO° 
units  of  time ;  therefore  it  revolves  through  —=-  In  1  unit 

360° 
of  time.     This  angular  velocity  of  —=-  per  unit  of  time 

can  be  resolved  into  two  components  in  direction  at  right 
angles  to  one  another ;  for  we  know,  from  dynamics,  that 
rotations  round  axes  can  be  resolved  in  exactly  the  same 
way  as  forces.  Therefore,  if  we  cut  off  ol  to  represent  an 

Q£JA 

angular  velocity  of  —=-  round  OP,  we  find  that  this  rotation 

is  equivalent  to  a  rotation  represented  by  Om  round  the 
radius  drawn  to  observer,  and  a  rotation  of  On  round  a  line 
at  right  angles  to  that  radius ;  but 

Om  =01  cos  (90  -  \]  =  01  sin  X  ; 

therefore  to  an  observer  at  A  the  plane  of  a  vibrating  pen- 

360° 
dulum  will  appear  to  revolve  through  — —  sin  X  in  1  unit  of 

time,  and  the  time  of  making  a  complete  revolution  would 
therefore  be 

SfiO  T 

=      *J  v      =  -±_  =  Tcosec  X  =  (23h  56m  4s)  cosec  X. 
<360  smX 

—  sinX 

Foucault's  Experiment. 

24.  Foucault  took  a  heavy  iron  ball  and  let  it  hang  from 
the  roof  of  the  Pantheon  by  means  of  a  wire  about  200  feet 
long.  A  circular  ridge  of  sand  was  placed  in  such  a  position 
that  at  every  swing  of  the  pendulum  a  pin  attached  to  the 
lower  part  of  the  ball  just  scraped  a  mark  in  the  sand.  The 
ball  was  then  drawn  aside  by  means  of  a  cord,  and  when  at 
rest  the  cord  was  burnt  off  so  that  the  pendulum  should 
swing  in  as  true  a  plane  as  possible. 

It  was  then  observed  that  the  marks  made  in  the  sand  at 
each  swing  did  not  coincide,  but  that  the  plane  of  the  pen- 
dulum seemed  to  be  slowly  turning  round  with  a  watch-hand 
rotation.  What  actually  happened,  however,  was  that  the 


CHAP,  ii.]  FOUCAULT'S  EXPERIMENT.  31 

whole  Pantheon,  together  with  the  observer  and  the  circular 
ridge  of  sand,  slowly  rotated  in  the  opposite  direction. 

The  wire  was  taken  of  this  great  length  (200  feet)  in 
order  that  the  pendulum  might  move  very  slowly,  thus 
meeting  with  very  small  resistance  from  the  air,  which 
enables  it  to  keep  up  its  motion  for  a  long  time.  The 
reason  a  long  wire  ensures  a  long  time  of  vibration  is  that 
the  time  of  vibration  is  proportional  to  the  square  root  of  the 
length  of  the  pendulum 


therefore  the  longer  the  pendulum  is  made  the  greater  is  the 
time  of  one  vibration. 

If  it  were  possible  to  keep  the  pendulum  vibrating  long 
enough  at  Paris  to  enable  its  plane  to  appear  to  make  a 
complete  revolution  the  time  taken  would  be  about  32  hours. 

We  can  account  for  this  phenomenon  on  no  other  sup- 
position than  that  the  earth  revolves  round  an  axis,  and  as 
the  plane  of  the  pendulum  does  not  seem  to  change  at  the 
equator  we  know  that  the  axis  of  revolution  of  the  earth 
must  be  perpendicular  to  the  equator. 

There  are  various  other  phenomena  which  can  be  ex- 
plained on  the  hypothesis  of  the  rotation  Qliha~eartk,-  such 

as   trade   winds  *mj_flprf.fl.iTi    nnnst-flnf-   nnrrAnfa  in  fh&jQP^"  j 

also  the  revolution  of  cyclones  which  in  the  southern  hemi- 
sphere move  with  a  watch-hand  rotation,  and  in  the  northern 
hemisphere  in  the  opposite  direction. 

EXAMPLES. 

1.  Find  the  lowest  latitude  at  which  it  is  possible  to  have  a  midnight  sun. 

Ana.  66°  32'  north  or  south. 

2.  What  circles  on  the  earth  correspond  to  the  latitudes  66°  32'  north  or 
south  ?  Ans.     The  arctic  or  antarctic  circles. 

3.  What  is  the  highest  latitude  north  or  south  at  which  it  is  possible  to  see 
the  sun  in  the  zenith  at  noon  ?  Ans.  23°  28'. 


32  THE  EARTH.  [CHAP.  II. 

4.  What  is  the  latitude  of  a  place  at  which  the  celestial  equator  and  horizon 
coincide  ?  Ans.  90°,  at  the  poles. 

5.  What  is  the  latitude  of  a  place  at  which  the  ecliptic  coincides  with  the 
horizon  ?  Ans.  66°  32'. 

6.  Why  is  the  sun  never  seen  in  the  zenith  at  Dublin  ? 

7.  If  a  pendulum  be  made  to  vibrate  at  a  place  whose  latitude  is  30°,  in 
what  period  of  time  will  the  plane  of  vibration  appear  to  make  a  complete  revo- 
lution ? 

Here  T=  (23h  56m)  cosec  30° 

=  (23h  56ra)  2 
=  47h  52«. 

8.  At  what  part  of  the  earth  would  a  body  have  no  deviation  towards  the 
east  when  let  drop  from  a  height  ?  Am.    At  the  poles. 

9.  If  a  person  travelling  eastward  go  round  the  world,  he  will  at  the  end  of 
his  journey  appear  to  have  gained  a  day.     On  the  other  hand,  if  he  travel  west- 
ward, he  will  appear  to  lose  a  day.    Explain  this. 

10.  How  far  should  a  man  travel  northwards  from  the  equator  in  order  that 
the  altitude  of  the  pole  might  become  10°  ?    Assume  the  radius  of  the  earth  to 
be  4000  miles  (J.  S.,  T.  C.  D.). 

2  x  3-14159  x  4000 
Here  —60—      -; 

•        iy,  2  x  8-14169  x  4000  x  10  B698. 

360 


CHAPTER  III. 

THE    OBSERVATORY. 

Astronomical  Clock. 

25.  WE  have  seen  in  Chapter  I.  that  the  apparent  uniform 
revolution  of  the  stars  round  the  celestial  pole  is  completed 
in  about  4  minutes  less  time  than  the  apparent  diurnal 
revolution  of  the  sun.  This  latter  interval  of  time  (more 
accurately  its  mean  value  throughout  the  year)  is  what  is 
taken  as  the  ordinary  day,  and  is  called  a  mean  solar  day.  It 
is  divided  into  24  mean  solar  hours. 

On  the  other  hand,  the  interval  of  time  taken  by  the 
fixed  stars  to  complete  a  revolution  round  the  pole  is  called 
a  sidereal  day.  The  sidereal  day  is  divided  into  24  sidereal 
hours,  which  are  reckoned  from  1  to  24,  therefore  we  have — 

24  sidereal  hours  =  23h  56m  mean  solar  time. 

The  astronomical  clock  is  regulated  so  as  to  mark  sidereal 
time,  and  as  the  sidereal  day  commences  when  the  first  point 
of  Aries  is  on  the  meridian,  the  clock  should  then  be  set  to 
mark  Oh  Om  0s.  It  will  then  indicate  the  sidereal  hours  up 
to  24  when  the  next  transit  occurs. 

Definition. — The  sidereal  time  at  any  instant  is  the 
interval  that  has  -elapsed  since  the  preceding  transit  of  the 
first  point  of  Aries  expressed  in  sidereal  hours,  minutes,  &c. 

As  the  right  ascensions  of  heavenly  bodies  are  measured 
eastwards  along  the  equator  from  the  first  point  of  Aries,  it 
follows  that  those  stars  which  have  a  small  right  ascension  will 
cross  the  meridian  before  those  stars  whose  right  ascensions 
are  greater.  In  fact,  360°  of  right  ascension  correspond  to 
24  sidereal  hours  or  15°  to  1  sidereal  hour.  Eight  ascensions 

T) 


34 


THE    OBSERVATORY. 


[CHAP.  in. 


may  therefore  be  expressed  in  degrees  or  in  time,  the 
former  being  reduced  to  the  latter  by  dividing  by  15.  Hence 
we  might  define  the  right  ascension  of  a  body  as  the  sidereal 
time  of  its  passage  across  the  meridian. 

The  hour  angle  of  the  first  point  of  Aries  (Art.  14)  at 
any  instant  reduced  to  time  (by  dividing  by  15)  is  evidently 
the  sidereal  time  at  that  instant. 

The  Transit  Instrument, 

26.  The  object  of  this  instrument  is  to  determine  the 
exact  instant  at  which  a  body  crosses  the  meridian.  It 
consists  of  a  telescope  rigidly  fixed  to  a  horizontal  axis.  At 
the  extremities  of  this  horizontal  axis  are  two  cylindrical 
pivots,  of  the  same  diameter,  which  move  in  sockets  fixed  on 
two  piers  of  solid  masonry.  In  order  to  diminish  the  pres- 
sure of  the  pivots  on  the  sockets, 
and  consequently  the  wear  caused 
by  friction,  a  great  part  of  the 
weight  of  the  telescope  is  balanced 
by  two  weights  which  are  hung  at 
the  extremities  of  a  pair  of  levers, 
the  other  extremities  being  attached 
to  the  cylindrical  pivots  (see  fig.  13). 

In  the  plane  of  the  principal 
focus  of  the  object-glass  is  placed 
a  framework  of  five,  or  seven, 
or  a  greater  number  of  vertical 
wires  or  spider  lines  (see  fig.  14) 
placed  at  equal  intervals  apart. 
These  are  intersected  at  right 
angles  by  two  horizontal  lines,  to  which  the  path  of  the  image 
of  a  star  or  other  body  across  the  field  of  view  will  be  almost 
parallel  and  in  a  position  midway  between  them.* 

*  In  some  portions  of  this  Chapter,  to  avoid  complexity,  mention  is  only 
made  of  one  horizontal  line  supposed  to  be  situated  midway  between  the  two 
mentioned  above. 


FIG.  13. 


CHAP..  III.] 


TRANSIT    INSTRUMENT. 


35 


FIG.  14. 


Since  the  principal  focus  of  the  object-glass  is  in  the 
same  plane  as  that  in  which  the  lines  are  stretched,  the  image 
of  a  star  under  observation  and  the  spider  lines  can  he  seen 
at  the  same  time,  and  for  purposes  of  adjustment  the  frame- 
work of  lines  admits  of  various  move- 
ments by  means  of  screws.  When 
the  instrument  is  used  at  night  it  is 
necessary  to  illuminate  the  spider 
lines.  This  is  done  by  means  of  a 
lamp  placed  opposite  one  of  the 
cylindrical  pivots,  the  light  from 
which  by  means  of  mirrors  is  re- 
flected down  the  tube  on  to  the  lines. 

It  is  the  object  of  the  observer  that  the  telescope  be  so 
adjusted  that  the  middle  vertical  spider  line  may  coincide  as 
nearly  as  possible  with  the  meridian.  The  time  at  which  the 
star  crosses  the  meridian  can  therefore  be  estimated  by  ob- 
serving the  instant,  as  indicated  by  the  astronomical  clock, 
at  which  it  crosses  this  line.  But  as  there  is  always  a  small 
error  in  noting  the  time  of  transit  over  one  line,  the  seven 
are  used  in  order  that  the  observer  may  note  the  time  at 
which  the  image  crosses  each  of  them,  when  the  mean  of  these 
being  taken  will,  in  all  probability,  give  a  more  accurate  result 
than  one  observation  could  afford,  as  the  observer  may  in  some 
cases  be  too  precipitate,  in  others  too  tardy,  the  positive  and 
negative  errors  thus  to  some  extent  neutralizing  one  another. 

Line  of  Confutation. —  When  the  image  of  an  object 
is  formed  at  the  principal  focus  of  the  object-glass  of  a 
telescope,  then  the  direction  in  which  it  is  viewed  is  the  same 
as  its  true  direction  as  seen  by  the  naked  eye  ;  this  line  along 
which  the  object  is  viewed  is  called  the  line  of  collimation,  or 
line  of  sight.  For  practical  purposes  the  line  of  collimation 
of  a  telescope  may  be  defined  as  the  line  joining  the  optical 
centre  of  the  object-glass  with  that  point  of  the  central 
vertical  spider  line  midway  between  the  two  horizontal  lines. 

T)  2 


36  THE   OBSERVATORY.  [CHAP.  III. 

Collimation,  Level,  and  Deviation  Errors,  with  the  corresponding 

Adjustments. 

27.  Every  transit  instrument,  to  be  perfectly  adjusted, 
must  satisfy  the  three  following  conditions  : — 

(1)  The  line  of  collimation  should  be  perpendicular  to 
the  axis  of  rotation  of  the  telescope. 

(2)  The  axis  of  rotation  should  be  horizontal. 

(3)  This  horizontal  axis  should  point  due  east  and  west, 
that  is,  the  line  of  collimation  should  be  due  north  and  south. 

Corresponding  to  the  above  conditions  we  have  there- 
fore in  every  instrument  three  errors — (1)  collimation  error ; 
(2)  level  error ;  (3)  deviation  error  ;  and  to  correct  these  we 
have  three  corresponding  adjustments. 

Collimation  Error* 

The  error  of  collimation  may  be  defined  as  the  amount 
by  which  the  angle  between  the  line  of  collimation  and  tha 
axis  of  revolution  of  the  telescope  falls  short  of  a  right  angle. 

Let  XT  (see  fig.   15) 
represent  the  axis  of  revo-        P  \ 
lution  of  the  telescope,  and  \ 

AB  the  line  of  collimation  \ 

supposed     not     at     right  •& 

angles  to  XT. 

Let  the  telescope  be 
pointed  at  an  object  or 
mark  on  the  earth  placed 
at  some  distance  away, 
and  let  P  be  a  point  of 
the  object  which  coincides 
with  the  middle  vertical 
spider  line.  The  telescope  Fm- 15- 

is  then  reversed  in  its  bearings,  the  right  hand  pivot  being 

*  Besides  the  above  method  of  correcting  the  collimation  error  two  other  methods, 
which  are  more  frequently  used  in  observatories,  are  given  in  Arts.  36,  36A. 


CHAP.  III.]  TRANSIT  INSTRUMENT.  37 

placed  in  the  left  socket,  and  vice  versa.  If  the  point  P  still 
coincide  with  the  central  spider  line,  there  is  no  collimation 
error.  If  not,  the  line  of  collimation,  on  the  telescope  being 
reversed,  occupies  the  position  A'B',  making  an  equal  angle 
with  the  perpendicular  on  the  other  side,  the  error  of  colli- 
mation being  half  the  angle  between  AB  and  A'B'. 

To  correct  for  this  error  the  spider  lines  must  be  moved 
by  means  of  screws  until  the  central  line  coincides  with  the 
same  point  before  and  after  reversing  the  telescope.  When 
this  adjustment  has  been  made  we  know  that  the  line  of 
collimation  sweeps  out  a  great  circle  in  the  heavens.  The 
object  of  the  next  two  adjustments  will  be  to  insure  that  this 
great  circle  shall  coincide  with  the  meridian. 

Level  Error. 

28.  This  error  is  due  to  the  axis  of  revolution  not  being 
horizontal.     To  correct  for  it  we  make  use  of  a  spirit-level 
which  is  long  enough  to  reach  from  one  extremity  of  the 
axis  to  the  other.     The  level  is  first  hung  on  to  the  axis  by 
means  of  hooks,  and  the  position  of  the  bubble  is  noted  by 
means  of  a  scale  attached.     The  level  is  then  reversed  and 
the  reading  of  the  scale  again  noted.     If  the  bubble  occupies 
the  same  position  as  before  there  is  no  level  error.     But  if 
not,  one  end  of  the  axis  must  be  raised  or  lowered  by  means 
of  a  screw  until  the  reading  of  the  bubble  is  the  mean  of  the 
two  former  readings. 

After  correcting  for  this  error  we  know  that  the  line  of 
collimation  not  only  describes  a  great  circle,  but  that  this 
circle  passes  through  the  zenith,  or  in  other  words,  is  a 
vertical  circle. 

Deviation  Error  or  Error  of  Azimuth. 

29.  This  error  is  due  to  the  line  of  collimation  not  pointing 
due  north  and  south.     The  middle  vertical  spider  line  will 
therefore  not  coincide  with  the  meridian,  but  with  a  vertical 


THE  OBSERVATORY. 


[CHAP.  111. 


such  as  ZX  (see  fig.  16).  The  error  is  detected  by  observing* 
the  interval  between  the  upper  and  lower  transits  of  a  circum- 
polar  star  (preferably  the 
pole  star),  and  again  the 
interval  between  its  lower 
and  upper  transits.  These 
two  intervals  should  be 
equal,  as  the  meridian  bi- 
sects the  circle  described  by 
the  star  round  the  pole. 
But  if  not  equal,  the  line 
of  collimation  cannot  coin- 
cide with  the  meridian,  the 
star  appearing  to  transit  at  FIG.  16 

m  and  n  (fig.  16). 

To  correct  for  this  error  one  extremity  of  the  axis  must 
be  moved  horizontally  by  means  of  a  screw,  until  the  two 
intervals  above  mentioned  are  identical. 

Observing  a  Transit.     Eye  and  Ear  Method. 

30.  As  the  apparent  motion  of  a  star  across  the  field  of 
view  is  greatly  magnified  by  the  telescope,  the  star  may 
appear  at  one  side  of  a  wire  or  spider  line  at  the  termination 
of  one  second,  and  at  the  other  side  before  the  end  of  the 
following  second.  An  expert  observer,  however,  can  esti- 
mate to  a  small  fraction  of  a  second  the  instant  of  crossing 
the  wire.  When  the  star  appears  in  the  field  of  view  he 
writes  down  the  hour  and  minute  from  the  clock,  and  then, 
without  again  looking  up  from  his  observation,  keeps  counting 
the  seconds  by  the  beats  of  the  clock.  The  relative  distances 
of  the  image  of  the  star  to  the  right  and  left  of  the  wire  at 
the  end  of  two  consecutive  seconds  enables  him  to  determine 
the  exact  time  of  its  passage  across  the  wire.  A  similar 
observation  is  made  for  each  of  the  seven  wires.  > 

This  method  is  called  the  "  eye  and  ear  method." 


CHAP.  III.]  MERIDIAN  CIRCLE.  39 

The  Chronograph. 

A  more  accurate  method  of  observing  a  transit  is  now 
coming  into  general  use,  by  means  of  the  chronograph.  The 
clock  is  so  arranged  that  at  every  beat  an  electric  circuit  is 
broken,  which  causes  a  dot  to  be  made  on  a  sheet  of  paper 
wrapped  round  a  uniformly-revolving  cylinder.  The  cylin- 
der, besides  revolving,  let  us  say  round  a  vertical  axis,  has 
a  slow  motion,  either  up  or  down  in  the  direction  of  its 
length,  so  that  the  dots  corresponding  to  seconds  of  time 
are  arranged  at  equal  intervals  in  a  spiral  or  corkscrew  curve 
on  the  cylinder. 

The  observer  at  the  instant  of  transit  across  a  wire, 
presses  a  button,  which  causes  a  dot  to  be  made  on  the 
cylinder  in  addition  to  those  caused  by  the  clock-beats.  The 
position  of  this  dot  relatively  to  the  two  dots  caused  by  the 
clock  immediately  before  and  after,  enables  him,  by  direct 
measurement,  to  determine  the  time  of  transit  to  a  very  small 
fraction  of  a  second. 

Meridian  Circle. 
31.  The  meridian  circle,  or,  as  it  sometimes  called,  the 


.  17. 

transit  circle,  consists  of  a  transit  instrument  MN  (fig.  17) 


40 


THE  OBSERVATORY. 


[CHAP.  in. 


such  as  we  have  already  described,  with  the  addition  of  a  pair 
of  graduated  circles  placed  one  on  either  side  of  the  telescope. 
These  circles  are  fixed  with  their  planes  at  right  angles  to 
the  horizontal  axis,  and  revolve  with  the  telescope.  The  rim 
of  each  circle  is  graduated  by  means  of  fine  lines  usually  into 
intervals  of  5'.  As  mechanical  subdivision  cannot  go  much 
further  than  this,  the  intermediate  minutes  and  seconds  are 
determined  by  means  of  a  microscope.  Usually,  however, 
six  microscopes  placed  at  equal  intervals  are  used,  each  of 
them  being  read  off  and  the  mean  of  them  all  taken.  These 
are  represented  in  fig.  17  by  the  letters  A,  B,  C,  &c.  In 
addition  to  these  there  is  a  microscope  of  low  magnifying 
power  called  the  Pointer,  which  is  used  to  read  off  the  degrees 
and  graduations  corresponding  to  the  intervals  of  5'.  These 
microscopes  are  all  fixed,  and  therefore,  as  the  circles  revolve, 
the  graduations  pass  across  the  field  of  view  of  each  of  them. 
The  pointer  microscope  should  read  zero  when  the  line  of 
collimation  points  to  the  zenith. 


Heading  Microscopes. 

In  the  focal  plane  of  the  object-glass  of  each  of  the  six 
microscopes  is  fixed  a  small  metal  scale  mn,  cut  into  fine 
notches  and  called  a  comb. 
This  scale  and  the  image  of  the 
graduations  on  the  circle  are 
both  seen  together  in  the  field 
of  view  as  represented  in  fig.  18. 
There  are  5  notches  to  each 
interval  ab  of  the  graduated 
circle,  therefore  each  notch  cor- 
responds to  1'.  A  small  aperture 
p  in  the  scale  is  placed  to  mark  FIO.  is. 

the  central  notch,  so  that  when  the  pointer  is  opposite  a 
graduation^  shall  coincide  with  one  also.     A  spider  line  xy, 


CHAP.  III.] 


READING    MICROSCOPE. 


41 


stretched  across  the  field  of  view,  can  be  moved  parallel  to 
itself  by  means  of  a  micrometer  screw,  the  head  of  which  is 
divided  into  60  equal  parts.  Five  turns  of  the  screw-head 
serve  to  bring  the  spider  line  through  an  interval  ab  of  the 
graduated  circle ;  therefore  we  have 

5  turns  screw-head  =  5'  of  graduated  circle  ; 
/.     1  turn  =  1' 


1  division 


_  1 " 


Now,  suppose  we  have  to  take  the  reading  of  the  circle 
in  any  position.  The  pointer  microscope  gives  the  reading 
roughly  in  intervals  of  5'.  The  odd  minutes  are  given  by  the 
number  of  notches  in  the  comb  from  the  central  notch  p  to 
the  preceding  graduation,  say  at  c.  The  number  of  divisions 
through  which  the  screw-head  must  be  turned  in  order  to 
bring  the  spider  line  into  coincidence  with  c  from  the  preced- 
ing notch  will  give  the  number  of  seconds. 

Error  of  Eccentricity  avoided  by  taking  the  mean  of  two  micro- 
scope readings  by  diametrically  opposite  points. 

32.  The  error  of  eccentricity  is  caused  by  the  graduated 
circle  revolving  round  some  poin^  not  its  centre.     This  is 


FIG.  19. 


entirely  eliminated  by  taking  the  mean  of  the  readings  at 


42  THE    OBSERVATORY.  [CHAP.  III. 

diametrically  opposite  points.  For  let  0  be  the  centre  of  the 
circle  (fig.  19),  0'  the  point  round  which  the  circle  revolves. 
Now  let  the  circle  revolve  round  0'  through  an  angle  0,  so 
that  the  line  AB  occupies  the  position  CD°9  then  the  angles  a 
and  /3  subtended  at  the  centre  by  the  arcs  AC  and  BD  will 
correspond  to  the  two  readings  at  opposite  sides.  We  have 
to  prove  that  the  mean  of  a  and  ]3  will  equal  0. 
By  Euclid  (i.  32)  we  have 


also  a  =  0  +  $ ; 

To  find  the  Zenith  Point  on  the  Meridian  Circle. 

33.  We  have  already  stated  that  the  pointer  should  read 
zero  when  the  line  of  collimation  points  to  the  zenith.  But 
the  mean  reading  of  the  six  microscopes  is  not  generally 
zero  at  the  same  time.  We  have,  therefore,  in  every  transit 
circle  to  find  the  zenith  point  or  the  reading  of  the  circles 
corresponding  to  the  zenith. 

In  order  to  find  this  point  the  telescope  is  directed  verti- 
cally downwards  to  a  basin  of  mercury  placed  beneath  it. 
It  is  then  moved  until  the  fixed  horizontal  wire  and  its 
image  reflected  by  the  mercury  are  perfectly  coincident. 
The  line  of  collimation  then  points  directly  to  the  nadir. 
The  reading  of  the  pointer  and  microscopes,  therefore,  gives 
the  nadir  point,  180°  from  which  is  the  zenith  point. 

Another  method  of  finding  the  zenith  point  is  by  observ- 
ing the  pole  star  (which  is  chosen  because  on  account  of  its 
slow  motion  it  remains  a  long  time  in  the  field  of  view),  and 
taking  the  reading  of  the  circle  when  the  star  appears  on 
the  horizontal  wire.  The  telescope  is  then  depressed  until 
the  image  of  the  star  reflected  from  the  surface  of  a  basin  of 


CHAP.  III.]  MERIDIAN    ZENITH    DISTANCE.  43 

mercury  coincides  with  the  horizontal  wire,  and  the  reading 
again  taken.  As  the  telescope  in  the  two  instances  must 
have  been  inclined  at  equal  angles  above  and  below  the 
horizon,  the  mean  of  the  two  readings  gives  the  horizontal 
point,  90°  from  which  is  the  zenith  point. 

It  is  also  evident  that  half  the  difference  of  these  two 
readings,  corresponding  to  the  direction  of  a  star  and  of  its 
image  in  a  basin  of  mercury,  is  the  meridian  altitude  of  the 
star. 

The  Polar  Point  on  the  Meridian  Circle. 

33A.  To  find  the  polar-point*  i.e.  the  reading  of  the 
meridian  circle  when  the  telescope  is  pointed  to  the  pole. 

A  circumpolar  star  is  observed  in  its  upper  and  lower 
transits,  and  in  each  case  the  reading  of  the  meridian  circle 
is  taken. 

z 


FIG.  19A. 

Thus,  if  m  and  n  represent  the  positions  of  the  star  at  its 
upper  and  lower  culminations,  we  have 

Pm  =  Pn  =  polar  distance  of  star  =  A. 

Let  polar  point  =  x. 

Then  x  +  A  =  reading  of  circle  when  star  is  at  n, 
and  x-  A  =  reading  of  circle  when  star  is  at  m. 
Therefore  the  polar  point  x  =  half  the  sum  of  the  two  readings. 


44  THE    OBSERVATORY.  [CHAP.  III. 

Since  PR  =  altitude  of  pole  =  latitude  of  place  ;  .*.  ZP  = 
the  complement  of  the  latitude  =  colatitude. 

The  latitude  of  the  observatory  or  place  can  now  be  found, 
for  the  difference  between  the  zenith  point  and  the  polar  point  is 
the  colatitude  ZP  which,  subtracted  from  90°,  gives  the  latitude. 


Meridian  Zenith  Distance.     Meridian  Altitude.     Declination. 

34.  In  order  to  measure  the  zenith  distance  of  a  star  when 
in  the  meridian,  the  telescope  is  pointed  to  the  star  and  the 
reading  of  the  circle  taken.  The  difference  between  this 
reading  and  the  zenith  reading  gives  the  meridian  zenith 
distance  of  the  body,  which  must  be  corrected  for  refrac- 
tion and  other  errors. 

The  meridian  altitude  is  obtained  by  subtracting  the 


Q 


FIG.  20. 


observed  zenith  distance  from  90°.  Having  obtained  the 
meridian  altitude  of  the  star  we  can  now,  the  latitude  of  the 
place  being  known,  determine  its  declination.  For  let  S  be  the 
position  of  a  star  in  the  meridian,  then  : — 

S&=  meridian  altitude  =  a ; 

SE  =  decimation  =  S. 


CHAP.  III.]  ASTRONOMICAL  CLOCK.  45 

Also.Z£5~  =  colatitude  ZP  (both  having  a  common  comple- 
ment ZE).    Now  we  have 


or     colat  +  S  =  a. 

Similarly,  if  the  star  be  at  S',  we  have 
colat-  S  =  a. 

Therefore  colatitude  ±  declination  =  meridian  altitude, 
the  plus  or  minus  sign  being  taken  in  the  northern  hemi- 
sphere, according  as  the  star's  declination  is  north  or  south.* 
From  this  equation,  knowing  the  meridian  altitude  of  the 
star  and  the  latitude  of  the  place,  its  declination  is  deter- 
mined. 

35.  Standard  Stars.  —In  the  Nautical  Almanac,  which 
is  published  every  year,  is  a  list  of  stars  whose  right  ascen- 
sions and  declinations  are  recorded  for  each  day.  Their 
declinations  are  determined  by  the  method  we  have  just  in- 
dicated, and  a  method  of  finding  their  right  ascensions  will 
be  given  later  on  (see  Flamsteed's  Method,  Chapter  VIII.)  . 
These  stars  are  called  standard  stars.  The  declination  of 
any  other  star  can  be  found  by  comparing  the  reading  of 
the  transit  circle  when  the  telescope  is  directed  to  the  star 
with  the  corresponding  reading  for  a  standard  star.  The 
difference  of  the  two  readings  being  the  difference  of  their 
declinations,  the  declination  of  the  body  in  question  can 
therefore  be  found. 

Regulation  of  the  Clock. 

As  the  first  point  of  Aries  is  an  imaginary  point  in  the 
sky,  such  that  we  cannot  observe  its  passage  across  the  meri- 
dian, we  are  not  therefore  able  to  tell  by  direct  observation 
when  to  set  the  clock  at  Oh  Om  0s.  But  the  time  of  transit  of 


*  In  case  the  star  transits  between  Z  and  P  the  equation  becomes 
colat  +  S  =  180  -  a. 


46  THE  OBSERVATORY.  [CHAP.  III. 

a  standard  star  is  known  from  its  right  ascension,  and  the 
clock  can  therefore  be  set  at  correct  sidereal  time  when  one 
of  these  stars  is  observed  in  the  meridian. 

The  rate  of  the  clock,  i.e.  the  amount  it  gains  or  loses 
each  day,  can  be  determined  by  noting  the  interval  between 
the  transits  of  a  fixed  star  on  two  successive  nights.  The 
interval  should  be  24  sidereal  hours,  from  which  the  daily 
gain  or  loss  of  the  clock  is  found.  A  good  clock  should  be 
such  that  its  rate  of  gain  or  loss  is  uniform. 

To  find  the  Right  Ascension  of  a  Body. 

The  clock  being  set  correctly  and  its  rate  being  known, 
then  the  sidereal  time  at  which  a  body  crosses  the  meridian 
is  its  right  ascension,  which  can  be  reduced  to  degrees, 
minutes,  and  seconds  by  multiplying  by  15. 

Collimating  Telescopes. 

36.  In  order  to  correct  the  error  of  collimation,  we  have 
seen  that  the  axis  of  the  telescope  has  to  be  reversed  in  its 
sockets,  and  the  direction  of  a  distant  mark  observed  before 
and  after  reversal.  This  method  has,  however,  now  been 
superseded  by  the  use  of  two  small  telescopes,  called  col- 
limating  telescopes.^  fixed  one  to  the  north,  and  the  other 
to  the  south  side  of  the  transit  telescope.  Each  of  these  is 
furnished  with  cross  wires,  so  that,  on  looking  through  one 
into  the  other,  which  can  be  managed  by  means  of  an 
opening  in  the  tube  of  the  large  telescope,  the  images  of  the 
cross  wires  (illuminated)  appear  coincident.  If  now  the  cross 
wires  of  the  transit  instrument  itself  be  so  adjusted  as  to 
coincide  with  those  of  the  north  collimator,  and  it  be  found 
on  rotating  the  telescope  round  that  they  are  also  coincident 
with  those  on  the  south,  the  line  of  collimation  must  be 
perpendicular  to  the  horizontal  axis.  By  this  method  the 
troublesome  operation  of  reversing  the  axis  of  the  telescope 
is  avoided. 


CHAP.  III.]:  EQUATORIAL.       MICROMETER.  47 

36A.  There  is  yet  another  method  of  determining  the 
collimation  error  by  pointing  the  telescope  vertically  down- 
wards towards  a  hasin  of  mercury.  If  the  axis  be  perfectly 
horizontal  and  there  be  no  collimation  error,  the  spider  lines 
should  coincide  with  their  image  formed  by  reflection  from 
the  surface  of  the  mercury  ;  for  the  rays  of  light  divergiug 
from  the  spider  lines  (which  are  illuminated),  after  passing 
through  the  object  glass,  fall  in  parallel  lines  on  the  surface 
of  the  mercury,  from  which  they  are  again  reflected  in 
parallel  lines,  which  are  converged  back  again  to  its  focus  by 
the  object  glass.  If,  therefore,  the  level  error  having  been 
previously  corrected,  the  real  system  of  spider  lines  do  not 
coincide  with  the  reflected  system,  the  difference,  which  may 
be  measured  by  a  micrometer,  is  twice  the  error  of  colli- 
mation. 

The  Equatorial. 

37.  Most  of  the  large  telescopes  in  observatories  are 
mounted  equatorially.  This  arrangement  consists  in  an 
axis  AB  (fig.  21)  which  points  to  the  celestial  pole,  called 


FIG.  21. 

the  polar  axis.     This  polar  axis  turns  in  fixed  bearings  A  and 
B  attached  to  two  fixed  piers.     The  telescope  can  be  turned 


48  THE  OBSERVATORY.  [CHAP.  III. 

round  an  axis  (7,  so  as  to  be  set  at  any  angle  to  the  polar 
axis.  A  clockwork  apparatus  is  generally  attached  to  the 
larger  instruments,  by  means  of  which  the  polar  axis  is  made 
to  revolve  uniformly  in  its  bearings  in  the  same  direction  as 
the  diurnal  motion  of  the  heavens,  the  revolution  being 
completed  in  23h  56m,  so  that  once  the  telescope  is  pointed 
at  a  star,  and  the  clockwork  apparatus  set  going,  it  is 
possible  to  keep  that  star  in  the  field  of  view  for  a  prolonged 
period. 

By  combining  the  two  motions  which  it  is  possible  to 
give  the  equatorial,  it  can  be  pointed  at  any  star  in  the 
heavens,  which  need  not,  as  in  the  case  of  the  transit  in- 
strument, be  in  the  meridian.  It  is  therefore  for  obser- 
vation of  bodies  not  in  the  meridian  that  this  instrument  is 
used. 

A  graduated  circle  mn,  whose  plane  is  at  right  angles  to 
the  polar  axis,  serves  to  set  the  telescope  at  any  required 
right  ascension.  It  is  called  the  hour  circle.  The  axis  C, 
round  which  the  telescope  turns,  also  carries  a  graduated 
circle,  which  is  not  drawn  -on  the  figure.  It  is  called  the 
declination  circle,  as  by  means  of  it  the  telescope  can  be  set 
at  any  required  declination.  Both  circles  are  read  off  by 
pointer  microscopes. 

The  equatorial,  on  account  of  its  high  magnifying  power, 
enables  us  to  observe  the  nature  of  the  moon  and  planets  and 
other  heavenly  bodies.  It  is  also  used  in  stellar  photography 
and  in  the  spectroscopic  analysis  of  the  stars. 

Micrometers. 

38.  Every  equatorial  is  furnished  with  a  micrometer  for 
measuring  small  angular  distances  such  as  the  angle  subtended 
at  the  observer  by  two  neighbouring  stars  in  the  field  of  view 
of  the  telescope.  The  kind  most  commonly  used  is  the 
parallel  wire  or  spider  line  micrometer.  It  consists  of  a 


CHAP.   III.]  MICROMETERS.  49 

rectangular  framework  (fig.   21  A)  with  a  graduated  screw- 


FlG.  2lA. 

head  gg  at  each  end ;  bbb,  ccc  are  two  metal  forks  which  slide 
within  one  another  on  which  are  fixed  two  parallel  spider 
lines  d  and  e ;  two  fine  screws/,/  having  milled  heads  g,  g 
connected  with  graduated  circles,  are  attached  one  to  each 
fork,  so  that  by  turning  these  milled  heads  each  fork  can  be 
drawn  out  or  pushed  in  according  to  the  direction  in  which 
the  head  is  turned,  and  thus  the  spider  lines  can  be  brought 
as  wide  apart  or  placed  as  close  together  as  we  please.  There  is 
also  a  fixed  transverse  spider  line  k  at  right  angles  to  d  and  e. 
The  circumference  of  each  of  the  circles  in  connexion  with 
the  screw- heads  is  divided  into  100  equal  parts.  There  is 
also  a  fine  scale  cut  into  notches,  every  fifth  notch  being  cut 
deeper  than  the  others  as  is  seen  in  the  above  diagram  ; 
the  distance  between  two  consecutive  teeth  being  equal  to* 
the  interval  between  the  threads  of  each  of  the  screws,  there- 
fore a  complete  revolution  of  one  of  the  screw-heads  just 
moves  the  corresponding  spider  line  through  a  distance  equal 
to  the  common  interval  between  the  teeth. 

In  order  to  measure  the  angle  subtended  by  two  neighbour- 
ing stars  at  the  observer,  the  micrometer  is  placed  in  the  focal 
plane  of  the  telescope  and  rotated  until  the  fixed  transverse 
spider  line  passes  through  the  images  of  the  two  stars.  The 
two  parallel  lines  are  then  shifted  by  means  of  the  screws 
until  each  coincides  with  an  image  of  a  star.  The  distance 
between  the  two  wires  can  now  be  found  by  noting,  by  means 
of  the  teeth  cut  in  the  scale,  how  many  turns  must  be  given 

E 


50  THE  OBSERVATOHY.  [CHAP.  III. 

to  the  screw-head  (or  screw-heads)  to  make  the  parallel  Hues 
coincide.  The  fractional  parts  of  a  turn  can  be  read  off  on 
the  graduated  circle  attached  to  each  screw-head ;  and  the 
angular  value  of  each  turn  being  known,  we  are  able  to  calcu- 
late the  angle  subtended  by  the  stars. 

The  micrometer  also  serves  to  measure  the  angular  dia- 
meters of  the  sun,  moon,  or  planets,  one  of  the  parallel  lines 
being  placed  so  as  to  touch  one  limb,  and  the  other  the  dia- 
metrically opposite  limb  of  the  circular  disc  presented  by  the 
body,  and  the  distance  between  them  is  measured  as  before. 

38A.  To  find  the  angular  value  of  each  turn  of  the  micrometer 
screw,  a  circumpolar  star  is  chosen,  preferably  the  pole  star, 
for,  on  account  of  its  very  small  distance  from  the  pole,  its 
motion  is  very  slow,  and  can  therefore  be  most  accurately 
observed.  The  micrometer  is  then  adjusted  so  that  the 
diurnal  motion  of  the  star  is  along  or  parallel  to  the  fixed 
spider  line.  The  two  movable  lines  are  then  separated  by  a 
certain  number  of  turns  of  the  screw,  and  the  time  taken  by 
the  image  of  the  star  to  pass  from  one  line  to  the  other  is 
noted,  from  which,  knowing  that  the  star  describes  360°  of  a 
small  circle  in  24  sidereal  hours,  the  angular  value  of  the 
distance  between  the  wires  is  easily  found,  and  hence  the 
angle,  expressed  in  seconds  of  a  small  circle,  corresponding 
to  one  turn  is  known  :  but  the  relative  magnitude  of  the 
small  circle  described  by  the  pole  star  to  a  great  circle  can  be 
found  since  the  declination  of  the  star  is  known  ;  hence  the 
number  of  seconds  of  the  arc  of  a  great  circle  corresponding 
to  each  turn  is  obtained. 

The  Alt-Azimuth  Instrument. 

The  alt-azimuth  may  be  described  as  an  equatorial,  of 
which  the  axis  points  to  the  zenith  instead  of  the  celestial 
pole.  It  admits  of  a  double  motion  in  altitude  and  azimuth, 
just  as  the  equatorial  does  in  right  ascension  and  declination. 
Like  the  equatorial,  it  is  used  in  ex-meridian  observations. 


€HAP.  III.]  EXAMPLES.  51 

Given  the  zenith  distances  of  a  circumpolar  star  at  its 
upper  and  lower  transits  to  calculate  the  latitude  of  the  place  and 
the  star's  declination. 

Let  the  zenith  distances*  Zn  and  Zm  (fig.  10A)  be  repre- 
sented by  %  and  2',  also 

Polar  distance  Pm  =  Pn  =  A,    and    ZP  =  colat ; 
.%     2  =  colat  +  A 
2'  =  colat  -  A  ; 

/.     2  +  2'  =  2  colat, 
and  2  -  2'  =  2  A ; 

,         2  +  2'     ,          .  .       .  A     s  +  2' 

.*.     colat  =  — jr —  ;  hence  lat.  =  90 — > 

£  2 

and  polar  distance  A  =  —^-  ;  hence  decln.  S  =  90 —  • 

<&  £ 

N.B. — If  the  star  in  one  of  its  transits  souths,  i.e.  if  it  cross 
the  meridian  south  of  the  zenith,  its  zenith  distance  at  this 
transit  is  to  be  considered  negative. 

EXAMPLES. 

1.  Supposing  the  earth  to  rotate  with  the  same  angular  velocity  as  at 
present,  but  in  the  opposite  direction,  what  would  be  the  length  of  a  mean  solar 
day  and  the  number  of  mean  solar  days  in  the  year  ?       Am.  23h  52m ;  367£. 

2.  How  many  sidereal  days  are  there  in  the  year  ?  Am.  366J. 

3.  What  is  the  meridian  altitude  of  the  sun  at  Dublin  on  the  21st  June,  the 
latitude  of  Dublin  being  53°  20'  ?  Ans.  60°  8'. 

Here  colat  ±8  =  0  (Art.  34) ; 

but  5  =  23°  28' N.     and    colat  =  90°  -  53°  20'  =  36°  40'  j 

.'.     36°  40'  +  23°  28'  =  o; 
.-.     a  =  60°  8'. 

4.  "What  is  the  meridian  altitude  of  the  sun  at  Dublin — (1)  during  the 
winter  solstice  ;  (2)  at  the  equinoxes  ?  Ans.  (1)  13°  12'. 

(2)  36°  40'. 
N.B.— At  winter  solstice  S  =  23°  28'  S.  (minus). 

*  In  all  cases  these  zenith  distances  when  measured  by  the  meridian  circle  are 
to  be  corrected  for  refraction  and  other  errors. 

E  2 


52  THE  OBSERVATORY.  [CHAP.  Ill- 

1/5.  The  zenith  distances  of  a  circumpolar  star  as  it  crosses  the  meridian  above 
and  helow  the  pole  are  found,  after  correcting  for  refraction,  &c.,  to  he  13°  7'  16" 
and  47°  18'  26".  Calculate  from  this  the  latitude  of  the  place  and  the  decli- 
nation of  the  star.  (See  Art.  38A).  Am.  59°  47'  9";  72°  54'  25". 

6.  The  latitude  of  John  o'  Groat's  house  is  58°  59'  N.     Find  the  sun's 
meridian  altitudes  at  that  place  on  midsummer  and  midwinter  days,  respectively. 

Ans.  54°  29';  7°  £3'. 

7.  Find  the  latitude  of  a  place  where  the  greatest  elevation  of  the  sun  above 
the  horizon  at  midsummer  is  76°  42'.  Ans.  36°  46'. 

^        8.  The  declination  of  Canopus  is  52°  38'  S. ;  if  we  travel  southwards,  where 
shall  we  first  find  it  attain  a  meridian  altitude  of  10°  ? 

Ans.  27°  22'  North  latitude. 

9.  Find  the  declination  of  a  star  whose  corrected  meridian  zenith  distance, 
as  observed  at  Dublin  (lat.  53°  20'),  is  72°- 18'  40".  Ans.  18°  58'  40"  S. 

10.  What  is  the  sun's   midnight  depression  below  the  horizon  at  Dublin 
during  midsummer  and  midwinter,  respectively.  Ans.  13°  12';  60°  8'. 

11 .  The  zenith  distances  of  a  star  at  lower  and  upper  culminations  are  found, 
after  correcting  for  refraction,  &c.,  to  be  76°  4'  and  2°  52°  S.  respectively. 

Find  the  latitude  of  the  place,  and  the  declination  of  the  star. 
N.B.—  Apply  formulae  in  Art.  38A,  but  2°  52'  being  south  is  given  a  minus 
sign.  Ans.  53°  24';  50°  32'. 

>V  12.  The  declination  of  Vega  (a  Lyrae)  is  38°  41'  N.  ;  does  it  cross  the 
meridian  of  Dublin  (lat.  53°  20')  north  or  south  of  the  zenith  ? 

Ans.  Upper  transit,  14°  39'  S.  of  zenith. 
Lower  transit,  87°  59'  N.  of  zenith. 


(    53    ) 


CHAPTER  IV. 

ATMOSPHERIC    REFRACTION. 

39.  WE  have  seen  (Chapter  in.)  how  the  altitude  of  a 
star  can  be  found  by  observation.  This  observed  altitude, 
however,  is  liable  to  some  error,  owing  to  the  rays  from  the 
star  being  bent  in  passing  through  the  atmosphere  before 
they  reach  the  eye  of  the  observer,  thus  leading  him  to  think 
tli at  the  star  is  in  a  different  direction  than  is  really  the  case. 
This  apparent  displacement  is  due  to  the  refracting  power  of 
the  atmosphere. 

We  know  from  op- 
tics that  when  a  ray 
of  light  passes  from  a 
rarer  to  a  denser  me- 
dium, it  is  refracted 
or  bent  towards  the 
perpendicular  to  the 
common  surface  of  the 
two  media.  Thus  A  OB 
would  represent  the 
path  of  such  a  refracted 
ray,  the  angle  i  being 
the  angle  of  incidence, 


r  the  angle  of  refrac- 


FIG.  22. 
tion,  while  i  -  r  is  the  amount  of  the  refraction. 

It  is  also  a  law  of  optics  that  the  angles  of  incidence  and 
refraction  are  such  that  their  sines  are  in  a  constant  ratio ; 
therefore, 

sin 


smr 


=  a  constant 


54 


ATMOSPHERIC    REFRACTION. 


[CHAP.  iv. 


Now  the  atmosphere  is  a  gaseous  fluid,  subject  to  the  action 
of  gravity.  Its  density  in  its  upper  layers  is  very  small ; 
but  as  we  approach  the  earth  its  density  increases  as  the 
weight  of  the  superincumbent  air  on  any  given  area  increases. 
Therefore,  when  a  ray  of  light  from  a  star  S  strikes  the 
atmosphere,  we  may  suppose  it  in  its  passage  to  the  earth  to 
pass  through  an  indefinite  number  of  media  each  denser  than 
the  preceding,  like  a  number  of  concentric  spherical  shells. 


The  path  AP  of  the  ray  through  the  atmosphere  thus  being 
continually  bent  will  be  curved.  To  an  observer  at  P  the 
star  will  appear  in  the  direction  PS',  a  tangent  to  the  curve 
at  the  point  P ;  whereas  its  real  direction,  if  there  were 
no  atmosphere  to  refract  the  ray,  would  be  PS,  a  parallel 
drawn  through  P  to  AS ;  for,  being  so  far  distant,  the  lines 
drawn  from  A  and  P  to  the  star  will  be  practically  parallel. 

The  angle  SPSf  between  the  apparent  direction  of  the 
star  and  the  direction  in  which  it  would  appear  if  there  were 
no  atmosphere  is  called  the  refraction. 

The  effect  of  refraction,  therefore,  on  the  position  of  a 
heavenly  body,  is  to  raise  it  in  the  sky,  so  as  to  increase 
its  altitude  and  diminish  its  zenith  distance.  But  as  this 


CHAP.  IV. J       VARIATION    IN    DENSITY    OF    ATMOSPHERE.  55 

apparent  displacement  takes  place  in  a  vertical  plane,  the 
azimuth  of  the  body  is  not  affected. 

Therefore,  in  all  observations  of  the  altitudes  of  heavenly 
bodies  each  apparent  altitude  must  be  diminished  by  the 
amount  of  the  refraction,  in  order  to  get  the  true  altitude. 
This  correction  is  called  the  correction  for  refraction. 

The  amount  of  refraction  is  greatest  when  the  angle  of 
incidence  is  greatest,  i.e.  when  the  body  is  on  the  horizon. 
The  refraction  is  then  called  the  horizontal  refraction. 

The  refraction  is  zero  at  the  zenith,  as  the  rays  from  a 
body  situated  right  overhead  strike  the  different  layers  of  the 
atmosphere  at  right  angles,  and,  therefore,  do  not  get  bent. 
The  horizontal  refraction  is  about  35'.  Therefore,  a  body 
on  the  horizon  will  appear  a  little  more  than  half  a  degree 
above  the  horizon.  As  the  angular  diameter  of  the  sun  is 
about  32',  or  a  little  more  than  half  a  degree,  we  are  able  to 
form  some  sort  of  idea  as  to  how  much  a  body  on  the  horizon 
is  displaced  by  refraction,  by  remembering  that  it  is  through 
an  arc  nearly  equal  to  the  breadth  of  the  sun's  disc.  From 
this  we  conclude  that  when  it  appears  to  us  that  the  sun  is 
about  to  set  he  has  in  reality  just  set,  and  we  would  not  see 
him  at  all  were  there  no  atmosphere  to  refract  his  rays. 

The  amount  of  refraction  is  influenced  by  the  changes  in 
the  pressure  and  temperature  of  the  atmosphere.  A  rise  in 
the  barometer  is  accompanied  by  an  increase  in  the  amount 
of  refraction,  provided  the  altitude  of  the  body  remain  the 
same.  On  the  contrary,  an  increase  of  temperature  produces 
a  diminution  of  refraction  under  the  same  circumstances. 
In  an  observatory  it  is  necessary,  in  estimating  the  error  due 
to  refraction,  to  take  into  account  not  only  the  zenith  dis- 
tance of  the  body,  but  also  the  pressure  and  temperature 
of  the  atmosphere  as  indicated  by  the  barometer  and  ther- 
mometer respectively.  We  have  seen  that  the  horizontal 
refraction  is  about  35' :  therefore,  how  rapidly  it  decreases  as 
the  zenith  distance  decreases  is  seen  from  the  fact  that  the 


56  ATMOSPHERIC  KEFRACTION.  [CHAP.  IV. 

refraction   at   an  altitude  of  45°  has  a  mean  value  of  only 
58"-2. 


Law  of  Refraction. 

40.  As  the  height  of  the  atmosphere  is  so  very  small 
compared  with  the  radius  of  the  earth,  we  may  assume  that 
the  lines  drawn  from  A  and  P  (fig.  23)  to  the  centre  of  the 
earth  are  parallel,  or,  in  other  words,  that  the  surface  of 
the  earth  is  a  horizontal  plane,  with  an  indefinite  number 
of  horizontal  layers  of  atmosphere  of  gradually  decreasing 
density  resting  on  it.  We  can  now  very  easily  deduce  a 
law  according  to  which  the  refraction  varies  ;  for  the  ray 
will  get  bent  through  the  same  amount  if,  instead  of  passing 
through  a  number  of  layers  of  varying  density,  we  suppose 
it  to  pass  through  a  homogeneous  atmosphere  of  the  same 
density  throughout  as  the  layer  in  contact  with  the  earth, 
when  we  can  imagine  it  to  get  bent  once  for  all  at  its 
entrance  into  the  atmosphere,  and  then  proceed  in  a  straight 
line  to  the  observer. 

The  refraction  of  a  heavenly  body,  the  temperature  and  pres- 
sure being  constant,  varies  as  the  tangent  of  the  apparent  zenith 
distance. 

Let  SAP  represent  the  path  of  a  ray  from  a  star  to  an 
observer  at  P  (fig.  24).  The  apparent  direction  of  a  star 
will  then  be  P/S",  the  angle  z  being  the  apparent,  and  z  +  x 
the  real  zenith  distance,  while  angle  SASf  =  amount  of 
refraction  =  x. 

,T  sine  (angle  of  incidence) 

Now  - — } — —. — j — 5 — -. — (  =  a  constant  =  p. 

sine  (angle  of  refraction) 

sin  (z  +  x]  ,       .      .  . 

or  -  =  p,  that  is,  sin  (z  +  x)  =  p  sin  z ; 

sin  z  cos  x  +  cos  z  sin  x  =  p  sin  z. 

But  x  is  a  very  small  angle,  and  we  know  from  trigo- 
nometry that  the  cosine  of  a  very  small  angle  is  almost  =  1, 


CHAP.  IV.]  LAW  OF  REFRACTION.  57 

and  as  the  perpendicular  and  arc  almost  coincide,  its  sine 
=  its  circular  measure  ; 

/.     sin  x  =  x  (expressed  in  circular  measure), 
and  cos  x  =  1 ; 


z    z 


Atmosphere 


Earth, 


FIG.  24. 

therefore  the  above  equation  becomes 

sin  z  +  x  cos  z  =  /n  sin  z  ; 
.*.     x  cos  z  =  fi  sin  z  -  sin  z  = 
.  sin  2 


or 


-  1)  sin  z  ; 


Let 


=  (/*  -  1)  tan  z. 

-  1  =  JT; 

=  IT  tan  z  ;  .*.  #  varies  as  tan  s. 


This  law  has  been  found  to  be  approximately  true  for 
zenith  distances  up  to  75°.  Nearer  the  horizon  the  law  does 
not  hold,  as  the  constitution  of  the  different  layers  of  the 
atmosphere  will  affect  it.  It  is  evident  at  once  that  the  law 


58 


ATMOSPHERIC  REFRACTION. 


[CHAP,  iv 


could  not  hold  at  the  horizon  where  the  zenith  distance  =  90°, 
for  tan  90°  =  infinity. 

The  amount  of  the  refraction  at  any  observed  zenith  dis- 
tance less  than  75°  can  be  found  by  substituting  for  tan  ^its 
value,  provided  the  value  of  the  constant  K  be  known,  for 
finding  which  we  give  the  following  methods. 

41.  To  find  the  constant  coefficient  of  refraction  when  the 
latitude  of  the  place  is  known. 

This  is  done  by  observing  with  the  meridian  circle  the 
zenith  distances  of  a  circumpolar  star  as  it  crosses  the 
meridian  above  and  below  the  pole. 

Let  m  and  n  repre- 
sent the  true  positions 
of  the  star  at  its  two 
culminations,  then  to 
the  observer  the  star 
will  appear  at  mf  and  n' 
as  raised  by  refraction. 
Let  the  observed  zenith 
distances  Zmf  and  Zn'\)Q 
represented  by  2  and  2' : 

FIG.  25, 

/.     Zm  =  Zm'  +  the  refraction  =  2  +  K  tan  2, 
Zn  =  Zn'  +  the  refraction  =  2'+  JTtan  2' ; 
adding,  we  get        Zm  +  Zn  =  2  +  si  +  K  (tan  2  +  tan  2'). 

But  Zm  +  Zn  =  Z  colat  (Art.  38A) 

(for  PR  =  latitude  of  place)  =  180°  -  2  lat ; 

/.     180°  -  2  lat  =  2  +  2'  +  K  (tan  2  +  tan  2') ; 

180°  -  2  lat  -  2- 2' _ 

tan  2  +  tan  2' 

But  the  latitude  of  the  place  is  known,  and  2  and  si  are 
the  observed  apparent  zenith  distances ;  therefore  K  is 
determined. 


CHAP,  iv.]  BRADLEY'S  METHOD.  59 

42.  Bradley's  "Method. — The  coefficient  of  refraction 
can  be  found  when  the  latitude  of  the  place  is  not  known 
by  the  method  of  Dr.  Bradley,  who,  besides  observing  the 
zenith  distances  of  a  circumpolar  star  at  its  two  culminations, 
measured  the  zenith  distances  of  the  sun  when  in  the  meri- 
dian at  the  summer  and  winter  solstices,  when  the  sun's 
declination  is  23°  28'  north  and  23°  28'  south  respectively. 
Let  these  observed  zenith  distances  be  denoted  by  s  and  /. 
If  now  A  and  B  (fig.  25)  represent  the  real  positions  of  the 
sun,  the  apparent  positions  when  observed  will  appear  raised 
to  A'  and  B' ; 

.*.     ZA  =  ZA'  +  refraction  =  s  +  K  tan  s, 

ZB  =-  ZB'  +  refraction  =  s'  +  K  tan  s' ; 
adding,  we  get 

ZA  +  ZB  =  s  +  s'  +  K  (tan  s  +  tans') ; 
but  ZA  +  ZB  =  2ZE  as  before,  and  arc  ZE  =  PR  (having  a 
common  complement  ZP)  =  lat ; 

.*.     2  lat  -  s  +  s'  +  K  (tan  s  +  tan  s'). 

We  have  also  from  observations  on  a  circumpolar  star,  as  in 
the  last  method, 

180°  -  2  lat  =  s  +  s'  +  K  (tan  s  +  tan  s'). 
By  adding  these  two  equations,  we  eliminate  the  latitude 
thus : — 

180°  =  s  +  s'  +  s  +  s'  +  K  {tan  s  +  tan  s'  +  tan  s  +  tan  s'j ; 
180°  -  s  -  s'  -  s  -  s' 


K 


tan  s  +  tan  s'  +  tan  s  +  tan  / 


But  s,  s',  s,  and  s'  are  observed;  therefore  K  is  deter- 
mined. The  fact  that  the  latitude  need  not  be  known  is  an 
advantage  in  Brad  ley's  method,  but  it  takes  six  months  to 
complete  the  observation. 

By  these  methods  the  constant  of  refraction  has  been 
estimated  at  about  58"-2  ;  /.  r  =  58"-2  tan  z. 


60  ATMOSPHERIC    REFRACTION.  [CHAP.  IV. 

42 A.  The  coefficient  of  refraction  may  also  be  found  and 
the  latitude  of  the  place  determined  at  the  same  time,  by 
observing  the  apparent  zenith  distances  of  two  circumpolar 
stars  at  their  transits  above  and  below  the  pole. 

Thus  we  have  for  one  star : — 

180°  -  2  lat  =  z  +  z'  +  K  (tan  z  +  tan  z') ; 
the  second  circumpolar  star  will  similarly  give 

180°  -  2  lat  =  2i  +  zi  +  K  (tan  zl  +  tan  2/) ; 
/.    z  +  z'  +  K  (tan  2  +  tan  2')  =  zl  +  Zi  +  K  (tan  %  +  tan  2/) ; 
.*.    K  (tan  2  +  tan  /  -  tan  21  -  tan  s/j  =  2i  +  2/  -  2  -  /, 

,        /  r 

—  Zi  -t  Zl     —  3  —  S 


tan  2  +  tan  2'  -  tan  21  -  tan 


•>  > 


the  value  of  K  being  thus  found,  the  latitude  may  be 
determined  by  substitution  in  one  of  the  above  equations. 
43.  A  curious  effect  of  refraction  is  the  oval  shapes  which 
the  sun  and  moon  appear  to  have  when  near  the  horizon. 
The  reason  of  this  phenomenon  is,  that  the  lower  limb  being 
nearer  the  horizon  than  the  upper  limb  will  be  raised  to 
a  greater  extent  by  refraction.  The  vertical  diameter  AB 


GD 


FIG.  26. 

will  therefore  appear  shortened  as  A'B',  while  the  horizontal 
diameter  remains  the  same.  This  apparent  diminution  in 
the  vertical  diameter  of  the  sun  and  moon  when  near  the 
horizon  amounts  to  about  one-sixth  part  of  the  whole,  or 
about  5'. 


CHAP.  IV.]  EXAMPLES.  61 


EXAMPLES. 

1  .  The  apparent  zenith  distance  of  a  starts  30°  :  calculate  the  true  zenith, 
assuming  the  coefficient  of  refraction  to  he  58"*2.J 
Here  the  refraction  =  58"  -2  tan  30°. 

=  58"-2x  _L  =  33"-6; 


.•.     True  zenith  distance  =  30°  0'  33"-6. 

2.  The  apparent  altitude  of  a  star  is  30°  ;  calculate  the  true  altitude,  the 
coefficient  of  refraction  heing  58"*2.  Ans.  29°  58'  19"-2. 

3.  An  altitude  of  a  star  is  observed,  and  found  to  he  the  angle  whose  sine 
is  1-3  ;  calculate  the  true  position  of  the  star,  assuming  the  amount  of  refraction 
at  an  altitude  of  45°  to  he  58"-2  (J.  S.,  T.C.D.). 

Here  the  refraction  =  jfiTtan  Z, 

hut    K=  58"-2  for  tan  45°  =  1  ,  and  tan  Z=  cot  (alt)  =  \*  ; 

.-.     refraction  =  ^x  58"-2  =  2'  19"-  7. 
Therefore  the  true  altitude  is  2'  19"  7  less  than  the  observed  altitude 

*'  4.  The  meridian  altitudes  of  a  circumpolar  star  are  20°  and  30°,  and  the 
corresponding  corrections  for  refraction  are  1'  40"  and  1'  9"  ;  find  the  latitude 
of  the  place  (Degree,  T.C.D.).  Ans.  24°  58'  35"-5. 

5.  If  a,  a'  he  the  true  and  apparent  altitudes  of  a  body  affected  by  refrac- 
tion, prove  the  equation  a  =  a!  —  58"  -2  cot  a'. 

'6.  Find  the  latitude  of  a  place  at  which  the  observed  meridian  zenith  distances 
of  a  circumpolar  star  were  47°  28'  and  22°  18',  given  that  the  tangents  of  these 
angles  are  1*09  and  *41  respectively,  and  taking  the  coefficient  of  refraction  to 
be  58"-2. 

Here  (Art.  41)     2  colat  =  Z  +  Z'  +  K  (tan  Z  +  tan  Z'}  ; 
or  2  colat  =  47°  28'  +  22°  18'  +  58"-2  (1-09  +  '41) 

=  69°  46'  +  58"-2  x  1-5 
=  69°  47'  27"-3  ; 
.-.  colat  =  34°  53'  43"-6  ; 
.-.  lat     =  55°    6'  16"-4. 


CHAPTER  V. 

THE   SUN. 

44.  To  an  inhabitant  of  the  earth  the  sun  is  by  far  the 
most  important  of  all  the  heavenly  bodies.  His  rays  supply 
light  and  heat  not  only  to  the  earth,  but  to  the  other  planets, 
and  his  attraction  controls  their  motions,  causing  them -to 
describe  their  respective  orbits.  It  is  therefore  hardly  to  be 
wondered  at,  that  from  the  most  ancient  times  a  body  of 
such  splendour,  whose  influence  on  earthly  affairs  was  so 
supreme,  should  have  been  an  object  of  great  awe  and 
veneration. 

The  sun  is  an  intensely  hot  and  luminous  body,  distant 
from  the  earth  by  about  92,700,000  miles.  The  angle 
which  the  diameter  of  his  disc  subtends  at  the  earth,  when 
measured  by  a  micrometer,  is  found  to  have  a  mean  value  of 
about  32'.  From  this  the  sun's  diameter  in  miles  can  be 
obtained,  for 

32'  x  60 <]__ 

206265"  "  92,700,000* 

Prom  which  we  get  d  the  diameter  of  the  sun  to  be  about 
860,000  miles,  or  about  110  times  the  earth's  diameter. 

As  the  volumes  of  two  spheres  are  to  one  another  as  the 
cubes  of  .their  diameters,  this  would  give 

vol.  of  sun  =  (110)3  x  vol.  of  earth 

=  1,331,000  x  vol.  of  earth. 

So  that  if  1,331,000  spheres   like  the   earth  were  massed 


CHAP.  V.]       DIURNAL  AND  ANNUAL  MOTIONS  OF  THE  SUN.        63 

together  into  one  sphere,  the  resulting  volume  would  about 
equal  that  of  the  sun. 

The  sun's  density,  however,  owing  to  his  physical  state, 
is  only  about  one-fourth  that  of  the  earth,  from  which  we 
conclude  that  his  mass  is  about  833,000  times  the  mass  of 
the  earth. 


The  Sun's  Apparent  Diurnal  and  Annual  Motions. 

45.  We  have  seen  in  Chapter  I.,  that  besides  a  compara- 
tively rapid  diurnal  motion  from  east  to  west  which  the  sun 
has  in  common  with  all  the  other  heavenly  bodies,  he  seems 
to  have  a  slow  motion  from  west  to  east  among  the  fixed 
stars  at  the  rate  of  about  1°  daily,  so  as  to  make  a  complete 
revolution  each  year.  A  mean  daily  change  in  right  ascen- 
sion of  1°  is  equivalent  to  4  minutes  of  time,  for  15°  corre- 
sponds to  one  hour.  The  mean  solar  day  is  thus  4  minutes 
longer  than  the  sidereal  day. 

We  have  seen,  in  Chapter  II.,  that  the  apparent  diurnal 
motion  of  the  heavenly  bodies  is  really  due  to  a  revolution 
of  the  earth  on  its  axis,  and  it  will  presently  be  shown  that 
the  sun's  apparent  annual  motion  in  the  ecliptic  is  due  to  a 
motion  of  the  earth  in  an  orbit  round  the  sun. 

On  account  of  the  sun's  change  of  position  among  the 
fixed  stars,  the  appearance  which  the  heavens  present  to  us 
each  night  at  a  certain  fixed  hour  goes  through  a  regular 
cycle  of  changes  in  the  course  of  the  year.  For  instance, 
stars  and  constellations  which  are  visible  at,  say  11  o'clock 
at  night  during  winter — such  as  Sirius,  Aldebaran,  the 
Pleiades,  and  the  constellation  of  Orion,  will  be  below  the 
horizon  at  the  same  hour  in  summer.  The  reason  of  this  is 
evident  if  it  is  borne  in  mind  that  11  P.M.  means  11  hours 
after  the  sun  has  been  in  the  meridian  ;  and  therefore,  when 
observed  at  the  same  hour  each  night,  each  star  will  have 
shifted  with  reference  to  both  meridian  and  horizon. 


64  THE  SUN,  [CHAP.  v. 

To  Trace  the  Annual  Path  of  the  Sun  on  the  Celestial 
Sphere. 

46.  On  account  of  the  sun's  brightness  it  is  impossible, 
even  in  an  observatory,  to  see  those  stars  which  are  at  all 
close  to  his  disc,  and  therefore  the  sun's  position  with  refe- 
rence to  them  cannot  be  directly  measured.  How,  then,  can 
the  ecliptic  be  traced  out  ?  To  the  ancient  astronomers,  who 
were  without  instruments  of  great  accuracy,  this  was  indeed 
a  difficult  problem.  Hipparchus  (160  B.C.)  noted  the  sun's 
position  relative  to  the  moon  during  the  daytime,  and  then, 
during  the  night,  he  determined  the  moon's  position  among 
the  fixed  stars,  from  which  he  deduced  the  position  occupied 
by  the  sun.  But  in  a  modern  observatory,  by  means  of  the 
transit  circle  and  astronomical  clock,  we  can  find  the  right 
ascension  and  declination  of  the  sun's  centre,  from  which  we 
are  able  to  note  on  the  celestial  globe  his  position  among  the 
fixed  stars.  By  repeating  these  observations  at  noon  each 
day,  his  annual  path  can  be  traced  out. 

When  the  ecliptic  is  thus  mapped  out  on  the  celestial 
sphere  it  is  found  to  be  a  great  circle,  that,  is,  its  plane  passes 
through  the  earth,  which  is  situated  at  its  centre.  But  let 
us  not  for  a  moment  suppose  that  the  sun's  apparent  yearly 
motion  could  be  explained  by  supposing  it  to  describe  a  circle 
round  the  earth  as  centre  merely  because  the  projection  of 
that  path  on  the  imaginary  celestial  sphere  is  a  circle.  For 
if  the  sun  were  to  move  in  a  circle  round  the  earth  as  centre, 
the  angle  subtended  by  the  diameter  of  its  disc  should  be 
always  the  same,  that  is,  supposing  that  the  sun  itself  does 
not  undergo  any  change  in  volume.  However,  we  find  that 
this  angle  is  not  constant,  but  goes  through  a  regular  cycle 
of  changes  throughout  the  year,  being  greatest  on  the  31st 
December,  when  it  is  32'  36",  and  least  on  1st  July,  when  it 
has  a  value  31'  32". 

From  this  it  is  seen  that  the  sun  is  nearest  the  earth  on 


CHAP,  v.]      EARTH'S  ANNUAL  MOTION.     PROOFS.  65 

31st  December,  and  furthest  away  on  1st  July,  but  that  the 
difference  is  not  very  great.  From  this  we  may  conclude 
that  if  the  sun  moves  round  the  earth,  his  path  must  be 
nearly,  but  not  quite,  circular. 

Apparent  Annual  Motion  of  the  Sun  due  to  a  Motion  of  the 

Earth. 

47.  As  the  apparent  annual  motion  of  the  sun  in  the 
ecliptic,  together  with  the  changes  in  the  seasons,  could  be 
explained  on  the  supposition  that  the  earth  describes  an 
annual  orbit  about  the  sun,  we  have  therefore  one  or  other 
of  two  alternatives  to  choose — 

Either  the  sun  revolves  round  the  earth  in  an  orbit  nearly 
circular ;  or 

The  earth  revolves  round  the  sun  in  an  orbit  nearly 
circular. 

That  the  second  explanation  is  the  only  admissible  one 
appears  from  the  following  considerations  : — 

(1)  It  is  known  (Chapter  VI.)  that  the  planets,  which 
are  opaque  bodies,  receiving  light  and  heat  from  the  sun  like 
the  earth,  revolve  round  the  sun  in  orbits  nearly  circular. 
That  some  of  these  are  much  larger  and  some  smaller  than 
the  earth ;  some  at  greater  and  some  at  less  distances  from 
the  sun;  also  the  earth's  periodic  time  (365|  days),  and 
its  mean  distance  from  the  sun    (92,000,000  miles)  satisfy 
Kepler's  3rd  Law  (Chapter  VI.),  viz.  that  the  squares  of 
the  periodic  times  of  the  planets  vary  as  the  cubes  of  their 
mean  distances  from  the  sun.      We  therefore  argue  from 
analogy  that   the   earth,  like   the  other   planets,   revolver 
round  the  sun. 

(2)  We  know  from  dynamical  principles  that  the  sun, 
earth,  and  planets,  on  account  of  their  mutual  attractions, 
must  either  come  together  or  revolve  round  the  common 
centre  of  gravity  of  the  whole  system.     But  the  sun's  mass 

F 


OF  THE 
I  W  I  \/  r  D  e  i  T  \/ 


66  THE  SUN.  [CHAP.  v. 

being  much  greater  than  that  of  all  the  planets  put  together, 
the  common  centre  of  gravity  of  all  is  a  point  within  the  sun 
not  far  removed  from  his  centre ;  and  round  this  point  the 
earth  and  planets  must  revolve. 

(3)  The  aberration  of  the  fixed  stars  (Chapter  VIII.) 
cannot  be  explained  on  any  other  hypothesis  except  on  the 
supposition  that  the  earth  moves  round  the  sun. 

Parallelism  of  the  Earth's  Axis. 

48.  We  find  that  although  the  earth  revolves  round  the 
sun,  the  position  of  the  celestial  pole  among  the  fixed  stars 
remains  very  nearly  constant  throughout  the  year  :  we  there- 
fore conclude  that  the  axis  of  the  earth  is  constant  in  direc- 
tion, i.e.  remains  parallel  to  itself,  while  the  earth  moves 
round  the  sun. 

Since  the  plane  of  the  ecliptic,  or,  in  other  words,  the 
plane  of  the  earth's  orbit,  is  inclined  to  the  equator  at  an 
angle  of  23°  28',  therefore  the  earth's  axis,  which  is  perpen- 
dicular to  the  equator,  must  be  inclined  to  the  plane  of  its 
orbit  at  an  angle  of  66°  32',  the  complement  of  23°  28'. 

The  Seasons. 

49.  The  changes  of  the  seasons  are  due   to   this   constant 
obliquity  of  the  earth's  axis  to  the  plane  of  its  orbit  (66°  32'). 

Let  fig.  27  represent  the  orbit  of  the  earth  round  the 
sun.  NS  represents  the  axis  of  the  earth,  of  which  there 
are  four  parallel  positions  taken  corresponding  to  the  summer 
and  winter  solstices,  and  the  autumnal  and  vernal  equi- 
noxes. EQ  represents  the  equator,  ab  and  cd  the  arctic  and 
antarctic  circles,  0  the  centre  of  the  earth,  and  H  the  sun. 

Position  (1)  Winter  Solstice  (left  side  of  figure}. 

This  represents  the  position  of  the  earth  when  the  north- 
ern portion  of  its  axis  is  turned  from  the  sun,  i.e.  when  the 
L  NOH  is  greatest,  which  happens  at  about  21st  December, 


CHAP.  V.] 


THE    SEASONS. 


67 


when  the  sun  is  vertical  to  the  Tropic  of  Capricorn  mn. 
Since  the  L  bOH  =  90°,  the  L  NOH  therefore  =  90°  +  23°  28' 
=  11 3°  28'.  To  an  observer  at  the  north  pole  -ZV'this  period 
coincides  with  the  middle  of  the  long  night  lasting  for  six 
months,  for  it  is  evident,  on  looking  at  the  figure,  that  the 
diurnal  revolution  of  the  earth  on  its  axis  could  not  bring 
any  part  not  distant  from  N  more  than  23°  28'  into  sunlight. 


(3) 


(2)       . 
FIG.  27. 

If  we  draw  a  small  circle  round  N  at  a  distance  from  it  of 
23°  28',  just  reaching  the  line  of  demarcation  of  light  and 
darkness,  this  circle  coincides  with  the  arctic  circle.  The 
reverse  is  the  case  round  the  south  pole,  where  this  period 
corresponds  to  the  middle  of  the  long  period  of  daylight. 
Similarly,  at  this  period  the  sun  will  not  set  even  at  12  o'clock, 
P.M.,  to  any  observer  within  the  antarctic  circle. 

Position  (3)  Summer  Solstice  (right  side  of  figure). 
Here  the  conditions  are  reversed :  the  north  pole  of  the 
earth  is  turned  towards  the  sun,  such  that  the  L  NOR  has 
its  least'value,  viz.  90°  -  23°  28'  =  66°  32'.  The  sun  in  this 
case  is  vertical  to  the  Tropic  of  Cancer  xij.  This  period  cor- 
responds to  the  middle  of  the  six  months'  daylight  at  the 
north  pole  and  six  months'  night  at  the  south  pole. 

F  2 


68  THE  SUN.  [CHAP,  v 

Positions  (2)  and  (4). 

These  two  positions  represent  the  earth  at  two  inter- 
mediate periods  when  the  plane  of  the  equator  passes  through 
the  sun,  which  therefore  occupies  a  position  on  the  celestial 
equator  at  one  or  other  of  the  equinoctial  points.  Here  the 
L  NOH  =  90° ;  therefore  the  line  of  demarcation  of  light  and 
darkness  will  pass  through  the  north  and  south  poles  of  the 
earth,  and  day  and  night  are  of  equal  duration  all  over  the 
world.  These  two  periods  are  therefore  called  the  two  equi- 
noxes, position  (2)  corresponding  to  the  vernal,  and  (4)  to 
the  autumnal  equinox. 

Amount  of  Heat  received  daily  from  the  Sun. 

50.  The  average  amount  of  heat  received  from  the  sun 
each  day  in  summer  is  greater  than  in  winter.  There  are 
two  reasons  for  this : — (1)  The  sun  remains  a  longer  time 
above  the  horizon  each  day  in  summer  than  in  winter  ;  and 
(2)  he  attains  a  greater  meridian  altitude  in  summer  than 
in  winter.  But  why  should  we  get  more  heat  from  the  sun 
when  he  has  a  great  meridian  altitude  than  when  he  is  low 


FIG.  28. 

down  near  the  horizon  ?  Could  the  explanation  be  that  he  is 
then  nearer  to  us? — No,  for  he  is  at  practically  the  same 
distance  from  us  at  noonday  when  his  rays  are  warm  as  at 
sunset,  when  he  seems  to  give  out  very  little  heat ;  and 


CHAP.  V.]  HEAT  FROM  THE  SUN.  69 

moreover  lie  is,  as  we  have  seen,  nearer  to  us  at  midwinter 
than  at  midsummer.  The  explanation,  however,  depends  on 
the  fact  that,  when  the  sun  has  a  great  altitude  in  the  sky, 
his  rays  strike  the  earth  directly ;  on  the  other  hand,  when 
low  down  near  the  horizon,  they  strike  obliquely.  Why  the 
efficiency  of  his  rays  in  warming  the  earth  should  be  greater 
in  the  former  case  than  in  the  latter  appears  at  once  from 
fig.  28.  Let  8  represent  a  point  on  the  sun,  SAB  and 
SAB'  two  cones  having  equal  vertical  angles  at  8,  the 
former  striking  the  earth  directly,  the  latter  obliquely,  so 
that  to  an  observer  on  the  earth  situated  inside  the  area  AB 
the  sun  will  appear  high  up  in  the  heavens,  and  viewed  from 
a  point  within  A'B'  he  will  appear  quite  close  to  the  horizon. 
We  may  now  assume,  since  the  cones  have  equal  vertical 
angles  that  equal  quantities  of  heat  radiate  from  8  along 
the  cones  SAB  and  SA'B',  and  that  therefore  the  areas  AB 
and  A'B'  receive  the  same  amount  of  heat,  but  the  area  A'B' 
being  an  oblique  section  of  the  cone  is  greater  than  AB, 
which  is  a  direct  section ;  therefore,  as  the  same  amount  is 
distributed  over  both,  the  quantity  of  heat  per  unit  of  area 
must  be  less  inside  A'B'  than  AB. 

This  explanation  accounts  both  for  the  fact  that  the  aver- 
age amount  of  heat  derived  from  the  sun  each  day  in  summer 
is  greater  than  in  winter,  and  also  that,  other  conditions 
being  the  same,  the  sun  should  feel  hotter  at  noon  on  any 
day  than  at  any  other  hour.  This  difference  is  still  further 
increased  owing  to  more  heat  being  absorbed  by  the  atmo- 
sphere when  the  sun  is  near  the  horizon,  for  the  rays  of  the 
sun  have  a  greater  thickness  of  atmosphere  to  pass  through 
when  coming  almost  horizontally  than  vertically. 

From  this  we  should  expect  that  in  northern  latitudes 
June  should  be  the  hottest  month  of  the  year,  and  December 
the  coldest.  But  we  generally  find  that  the  mean  tempera- 
ture is  higher  in  August  than  in  June,  and  lower  in  February 
than  in  December.  The  reason  of  this  is,  that  during  June 


70  THE  SUN.  [CHAP.  v. 

the  earth  has  not  had  sufficient  time  to  regain  the  heat  lost 
during  the  winter ;  but  for  some  months  after  June,  the 
earth  gains  more  heat  during  the  day  than  it  loses  at  night ; 
there  is,  therefore,  a  continuous  increase  in  the  mean  tempera- 
ture until  the  amounts  of  heat  gained  and  lost  during  the 
twenty-four  hours  become  equal.  Again,  for  some  time 
after  the  winter  solstice,  the  amount  of  heat  lost  during  the 
night  exceeds  that  gained  during  the  day  ;  therefore,  during 
this  period  the  earth  is  losing  heat,  the  lowest  mean  tem- 
perature being  in  general  registered  when  the  gain  and  loss 
during  the  twenty-four  hours  become  exactly  equal.  This  is 
the  explanation  of  the  old  saying  :  "  as  the  day  lengthens 
the  cold  strengthens." 

For  the  same  reason,  mid-day  is  not  generally  the 
warmest  hour  of  the  day,  as  there  is  in  general  a  continuous 
gain  in  heat  for  some  time  into  the  afternoon  :  nor  is  the 
coldest  period  of  the  night  generally  reached  for  some  hours 
after  midnight. 

The  mean  temperature  at  any  place  is,  however,  greatly 
influenced  by  other  conditions,  such  as  prevailing  winds, 
insular  or  continental  position,  proximity  to  the  gulf-stream, 
height  above  sea-level,  &c. 

Rotation  of  Sun.     Sun  Spots. 

51.  When  the  disc  of  the  sun  is  observed  through  a 
telescope,  dark  spots  are  very  often  seen  on  its  surface. 
These  appear  first  at  the  eastern  edge,  move  slowly  across 
the  bright  face  of  the  sun,  and  after  disappearing  behind  the 
western  edge,  reappear  again  on  the  same  side  as  before. 
Moreover,  the  times  of  appearance  and  disappearance  are 
equal,  each  being  about  13  J  days.  From  these  observations 
we  are  led  to  one  or  other  of  two  conclusions — 

(1)  Either  they  are  due  to  bodies  revolving  round  the 
sun,  so  that  they,  coming  between  the  sun  and  observer, 
appear  as  dark  spots  projected  on  the  sun's  surface ;  or 


CHAP.  V.]  ROTATION  OF  SUN.       SUN  SPOTS.  71 

(2)  They  are  due  to  actual  appearances  on  the  surface 
of  the  sun  itself,  the  sun  rotating  round  an  axis. 

That  the  first  conclusion  is  in  the  highest  degree  improb- 
able appears  at  once.  For  let  FCD  (fig.  29)  represent  the 
supposed  orbit  of  such  a  body  round  the  sun. 


D  J  ^      r 

5 

FIG.  29. 

AP  and  BP  are  tangents  drawn  to  the  sun  from  the 
observer  on  the  earth;  these  are  almost  parallel,  as  the 
observer  is  so  far  distant  compared  with  the  diameter  of  the 
sun.  Now  it  is  evident  that  the  time  during  which  a  body 
moving  in  the  orbit  FCD  would  appear  on  the  sun's  surface 
would  be  while  passing  through  the  arc  CD,  the  time  of 
disappearance  corresponding  to  the  arc  CFD ;  therefore, 
assuming  that  the  velocity  of  the  body  is  uniform,  the  time 
of  appearance  would  be  much  less  than  that  of  disappearance. 
But  observation  shows  that  these  two  periods  are  almost 
equal.  From  this  we  conclude  that  the  sun  rotates  round 
an  axis.  The  period  of  rotation  is,  however,  somewhat  less 
than  the  apparent  period  of  revolution  of  the  spots,  as  allow- 
ance must  be  made  for  the  motion  of  the  earth  in  its  orbit. 
The  period  for  the  spots  is  about  27  days,  while  the  sun 
rotates  once  in  25 1  days0 

These  spots  are  darker  at  the  centre  than  round  the 
margins.  The  dark  central  portion  is  called  the  umbra, 
surrounding  which  is  the  penumbra,  of  a  somewhat  lighter 
hue,  apparently  composed  of  radiating  filaments.  Apart 


72  THE  SUN.  [CHAP,  v 

altogether  from  the  motion  due  to  the  sun's  rotation  they  are 
observed  to  undergo  changes  in  their  size  and  shape,  and  after 
some  weeks  or  months  to  disappear  altogether.  "  The  infe- 
rence from  these  various  facts  is  irresistible."  (I  here  quote 
Sir  Bobert  Ball.)  "  It  tells  us  that  the  visible  surface  of  the 
sun  is  not  a  solid  mass — is  not  even  a  liquid  mass — but  that 
the  sun,  as  far  as  we  can  see  it,  consists  of  matter  in  the 
gaseous  or  vaporous  condition." 

"  It  often  happens  that  a  large  spot  divides  into  two  or 
more  smaller  spots,  and  these  parts  have  been  sometimes 
seen  to  fly  apart  with  a  velocity,  in  some  cases,  of  not  less 
than  1000  miles  an  hour." 

In  the  case  of  some  of  the  largest  spots  the  umbra  has 
been  found  to  subtend  an  angle  of  1'  30"  at  the  eye  of  the 
observer,  which  would  give  a  diameter  of  about  40,000 
miles — about  five  times  greater  than  that  of  the  earth. 

The  Sun  a  Sphere. 

52.  We  have  seen  that  the  sun  rotates  round  an  axis; 
we  also  know  that  the  shape  of  the  disc  which  he  turns  to 
the  observer  is  always  circular,  for  all  the  diameters,  when 
measured  in   different  directions    with  a  micrometer,    are 
found  to  be  equal.     The  sun  must  therefore  be  a  sphere,  as 
no  body  rotating  in  the  same  way  as  the  sun  does  could 
always  present  a  circular  margin  unless  it  were  spherical. 

Twilight. 

53.  After  sunset  a  considerable  time  elapses  before  com- 
plete darkness  sets  in.     We  call  this  interval  twilight.     There 
is  a  corresponding  interval  before  sunrise,  which  we  call  dawn. 

Twilight  is  caused  by  the  diffused  reflection  of  the  sun's 
rays  from  the  upper  layers  of  the  atmosphere.  After  the  sun 
sets,  when  his  rays,  on  account  of  the  curvature  of  the  earth, 
can  no  longer  reach  us,  he  still  continues  to  illumine  the 
atmosphere  or  particles  suspended  therein,  which  reflect  the 
light  down  to  us. 


CHAP.  V.]  TWILIGHT.  73 

Twilight  is  considered  at  an  end  when  minute  stars  of 
the  sixth  magnitude  can  be  seen  in  the  zenith.  Of  course, 
atmospheric  conditions  will  alter  considerably  the  interval  of 
time  after  sunset  which  must  elapse  before  this  takes  place  ; 
but  generally  stars  of  the  sixth  magnitude  appear  when  the 
perpendicular  distance  of  the  sun  below  the  horizon  exceeds 
18°.  Therefore,  twilight  lasts  until  the  perpendicular  distance 
of  the  sun  below  the  horizon  exceeds  18°. 

Twilight  is  shortest  at  the  equator.  The  student  will 
see  this  at  once  by  referring  to  the  diagram  of  the  celestial 
sphere  for  an  observer  at  the  equator  (Art.  20).  The  sun's 
diurnal  path  here  cuts  the  horizon  at  right  angles,  and  there- 
fore he  takes  a  very  short  time  to  get  18°  below  the  horizon. 
On  the  other  hand,  for  an  observer  in  the  British  Isles,  the 
sun  in  setting  cuts  the  horizon  at  an  acute  angle  (which  gets 
more  acute  the  further  north  we  go),  and  therefore  he  has  to 
skim  below  the  horizon  a  much  greater  distance,  and  for  a 
much  longer  time,  before  his  perpendicular  distance  below  the 
horizon  reaches  18°. 

54.  Twilight  at  the  North  and  South  Poles.— At 

the  north  pole  we  have  seen  (Chap.  II.)  that  the  sun  remains 
for  about  six  months  below  the  horizon,  from  23rd  of  Sep- 
tember until  21st  of  the  following  March.  He  is  never, 
however,  during  that  period,  at  a  very  great  perpendicular 
distance  below  the  horizon,  the  greatest  depth  being  23°  28', 
which  he  reaches  on  21st  December.  However,  of  this  six 
months  of  so-called  night  a  great  portion  is  twilight,  for  it 
will  not  be  altogether  dark  as  long  as  the  sun  is  within  18° 
of  the  horizon.  That  the  period  during  which  twilight  lasts 
will  be  as  great  a  portion  of  the  six  months  as  18°  is  of 
23°  28'  we  can  by  no  means  say,  for  the  sun's  change  in 
declination  is  not  uniform.  Assuming,  however,  that  this 
is  the  case,  we  would  have  about  four  out  of  the  six  months 
during  which  twilight  lasts,  viz.  two  months  after  the  23rd 


74 


THE    SUN. 


[CHAP,  v 


September  and  two  months  before  the  21st  of  the  following 
March. 

Of  course,  the  above  will  also  apply  to  the  south  pole 
during  the  period  when  the  sun  is  below  the  horizon,  viz. 
from  21st  March  till  the  23rd  September. 


To  find  the  duration  of  Twilight  at  the  Equator  during  the 
Equinoxes. 

55.  During  the  Equinoxes  the  sun's  diurnal  path  almost 
coincides  with  the  celes- 
tial equator,  which,  for 
an  observer  at  the  earth's 
equator,  cuts  the  horizon 
at  right  angles,  passing 
through  the  zenith  and 
nadir.  Let  S  represent 
the  sun  at  sunset  (fig.  30). 
Let  $'  represent  the  sun 
at  end  of  twilight. 

We  have  therefore  to 
find  the  interval  of  time 


POLE 


N 

FIG.  30. 
corresponding  to  SS'  or  18°  of  his  daily  course. 


But  360°  correspond  to  24  hours ; 

.'.     As  360°  :  18° : :  24h  :  x, 


18x24 
360 


ours  =  lh  12m. 


56.  To  calculate  the  duration  of  twilight  at  any  place 
we  have  to  solve  two  spherical 'triangles,  the  three  sides 
being  given.  . 

For,  let  S  represent  the  sun  at  sunset,  let  S'  represent  the 
sun  at  end  of  twilight.  Join  8  and  S'  to  zenith  and  pole 
by  four  arcs  of  great  circles. 


CHAP.  V.]          TO    FIND    DURATION    OF    TWILIGHT.  75 

Now,  the  sides  of  the  A  ZPSf  are  known ;  for  ZP  =  90° 
-  lat   =   colatitude;    ZS' 
=  90°  +  18°  =  108°,  since 
S'  is  18°  below  horizon,  E 


-  declination  of  sun.  But 
the  declination  of  the 
sun  is  known  for  each 
day  in  the  year  from  the 
Nautical  Almanac ;  there- 
fore P/S'  is  known  ;  there- 
fore by  solving  we  are 
able  to  calculate  the  Fl<>. 31- 

L  ZPS',   which  is  the  hour  angle   of   the  sun   at   end  of 
twilight. 

Similarly,  the  sides  of  the  A  ZPS  are  known,  and  there- 
fore we  can  solve  for  the  L  ZPS,  which  is  the  hour  angle  of 
the  sun  when  setting.  Subtracting  these  two  angles,  we  get 
the  t_SPS',  which  measures  the  duration  of  twilight.  Con- 
verting this  into  time  at  the  rate  of  360°  to  24  hours,  or  15° 
to  1  hour,  gives  the  duration. 

It  is  obvious  that  the  duration  of  twilight  depends  upon 
the  latitude  of  the  place  and  the  declination  of  the  sun,  for  these 
quantities  alone  are  involved  in  the  solution  of  the  above 
spherical  triangles.  That  is,  it  depends  on  the  part  of  the 
earth  at  which  the  observer  is  situated,  and,  even  in  the  same 
place,  it  varies  according  to  the  season  of  the  year. 

57.  It  is  evident  that  twilight  cannot  last  all  night  at  or 
near  the  equator,  the  sun's  diurnal  path  cutting  the  horizon 
at  nearly  a  right  angle.  The  question  then  arises  as  to 
what  are  the  conditions  which  must  hold,  in  order  that 
twilight  may  last  all  night : — 

Twilight  will  last  all  night  at  any  place,  provided  the 


76  THE  SUN.  [CHAP.  v. 

latitude  of  the  place  plus  the  declination  of  the  sun  is  not  less 
than  72°.  z 

For  let  8  represent  the 
sun  at  midnight  when  in        E 
meridian  below  the  horizon, 
then 

PE  =  alt.  of  pole  ==  lat'  H 
of  place  =  /,  and 

8Q  =  decl.  of  sun  =  S. 
Now        PQ  =  90°; 

that  is, 

ono  FIG.  32. 


But  if  twilight  just  lasts  all  night,  SR  =  18°  at  mid- 
night ; 


.-.     if  /+<Hs  =  or>  72°, 

twilight  lasts  all  night. 

This  rule  holds  when  the  latitude  of  the  place  and  the 
declination  of  the  sun  are  both  north  or  both  south.  When 
the  latitude  is  north  and  the  declination  of  the  sun  south,  or 
vice  versa,  then  the  condition  becomes 

/-  8  not  <  72°0 
EXAMPLES. 

1.  What  effect  would  be  produced  upon  the  seasons  if  the  earth's  axis  were 
in  the  plane  of  the  ecliptic  or  were  perpendicular  to  it  ? 

2.  If  the  declination  of  the  sun  be  10°,  find  the  lowest  latitude  at  which 
twilight  lasts  all  night. 

Here          I  +  8  =  72°, 

or    J+10°=72°; 

.-.    I  =  62°. 


CHAP.  V.]  EXAMPLES.  77 

3.  Find  the  latitude  of  the  place  for  which  twilight  just  lasts  all  night  when 
the  sun's  declination  is  16°  N.     (Degree  Exam.,  T.C.D.).  Am.  56°  N. 

4.  How  does  the  duration  of  twilight  at  a  given  place  alter  with  the  season 
of  the  year  ?  (S.  S.,  T.C.D.).  Am.     See  Art.  56. 

5.  Determine  the  limits  of  the  latitudes  of  places  at  which  twilight  lasts  all 
night  long,  when  the  sun's  declination  is  10°  15'  N. 

Ans.     At  lat.  61°  45'  and  places  further  north. 

6.  Find  the  decimation  of  the  sun  when  twilight  begins  to  last  all  night  at 
Dublin  (lat.  53°  20').  Ans.  18°  40'  N. 

7.  Find  the  lowest  latitude  at  which  it  is  possible  for  twilight  to  last  all 
night.  Am.  48°  32'. 

8.  Upon  what  does  the  duration  of  twilight  depend  ? 

Ans.  The  latitude  of  the  place  and  the 
declination  of  the  sun. 

9.  Can  twilight  last  all  night  at  Paris  (lat.  48°  50')  ?     (See  question  7.) 

Ans.  Yes,  but  only  for  several  nights  before 
and  after  the  summer  solstice. 

10.  Show  how,  by  solving  a  spherical  triangle,  the  time  of  sunset  or  sunrise 
can  be  calculated  for  any  place  at  a  given  date. 

Ans.  See  Art.  56. 


78 


CHAPTER  VI. 

THE  MOTIONS  OF  THE  PLANETS.      THE  SOLAR  SYSTEM. 

58.  WE  mentioned  in  a  previous  chapter  that  those 
planets  which  are  visible  to  the  naked  eye  and  with  which 
the  ancients,  who  were  not  possessed  of  telescopes,  were 
acquainted,  are; — Mercury,  Venus,  Mars,  Jupiter,  and 
Saturn. 

If  the  ordinary  observer  wish  to  find  out  whether  a 
bright  object  in  the  sky  be  a  planet  or  a  fixed  star  he  has 
only  to  note  its  position  with  reference  to  the  neighbouring 
fixed  stars ;  for  instance,  it  may  happen  to  be  in  a  line  with 
two  stars,  or  form  an  equilateral  triangle  with  them.  If, 
after  several  weeks,  the  body  seems  to  have  altered  its  posi- 
tion with  reference  to  these  fixed  stars,  it  is  probably  one  of 
the  above  planets. 

Since  the  invention  of  telescopes  several  other  large 
planets,  with  some  hundreds  of  very  small  ones,  have  been 
discovered.  The  names  of  the  planets  at  present  known  are, 
in  their  order,  from  the  sun  outwards — 


-^    ,,        i  Interior  Planets. 
Jbjartn 


Inferior  Planets    I     Mercury  \ 

Venus      ( 

Mars        / 

The  Asteroids 

Superior  Planets   <!  | 

Saturn  . 

-rr  }•  Exterior  Planets. 

Uranus 

Neptune  J 


CHAP.  VI.] 


DEFINITIONS. 


79- 


Those  planets  whose  orbits  lie  between  the  earth  and  the 
sun  are  called  inferior  planets.  Those  whose  orbits  lie  out- 
side that  of  the  earth  are  called  superior  planets.  Thus 
Mercury  and  Yenus  are  inferior  planets,  whereas  Mars, 
Jupiter,  &c.,  are  superior  planets. 

There  are  also  interior  and  exterior  planets — those  whose 
orbits  lie  between  the  Asteroids  and  the  sun  being  interior, 
and  those  outside  exterior. 


The  orbits  of  the  Planets  cut  the  Plane  of  the  Ecliptic  at 
very  small  angles. 

59.  It  is  to  be  observed  that,  throughout  the  whole  of  a 
planet's  path  round  the  sun,  it  never  appears  more  than 
a  few  degrees  above  or  below  the  ecliptic.  The  conclusion 
from  this  is  obvious — that  the  orbits  of  the  planets  round 
the  sun  are  nearly  in  the  plane  of  the  ecliptic,  i.e.  the  plane 
of  the  earth's  orbit ;  in  fact  they  cut  the  plane  of  the  ecliptic 
at  very  small  angles. 

Definition. — A  planet  is  said  to  be  in  inferior  conjunc- 
tion when  it  comes  between  , 
the  earth  and  the  sun,  and 
in  superior  conjunctionwhen 
the  sun  is  between  the 
earth  and  the  planet.  Thus 
if  E  represents  the  earth, 
V  will  be  the  position  of  a 
planet  in  inferior  conjunc- 
tion, and  V  in  superior 
conjunction  (fig.  33). 

A   planet   is    said   to 

be  in  opposition  when  the 

i  ^ 

earth  comes  between  the 

sun  and  the  planet.     Thus  M  represents  the  position  of  a 
planet  in  opposition. 


80  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.    VI. 

I  It  is  evident  that  only  an  inferior  planet  can  be  in  inferior 
conjunction,  and  only  a  superior  planet  in  opposition. 

By  the  nodes  of  a  planet  are  meant  the  two  points  where 
its  orbit  cuts  the  plane  of  the  ecliptic,  i.e.  the  plane  of  the 
earth's  orbit.  That  point  through  which  the  planet  passes 
in  going  from  the  southern  to  the  northern  side  of  the 
ecliptic  is  called  the  ascending  node,  the  other  the  descending 
node. 

N.B. — It  is  evident  that  if  the  orbits  of  the  planets  were 
in  the  plane  of  the  earth's  orbit  instead  of  cutting  it  at  small 
angles,  it  would  be  possible  to  see  the  transit  of  an  inferior 
planet  across  the  sun's  disc  each  time  inferior  conjunction 
occurs.  But  this  phenomenon  is  of  very  rare  occurrence  ;  for 
although  the  inferior  planets  are  very  often  in  inferior  con- 
junction, they,  not  being  at  the  same  time  in  the  plane  of  the 
earth's  orbit,  will  be  situated  above  or  below  the  sun ;  and 
when  these  planets  are  in  the  plane  of  the  earth's  orbit, 
which  occurs  when  they  are  passing  through  their  nodes, 
they  do  not  happen  to  be  in  inferior  conjunction.  Therefore, 
the  planet  must  be  in  conjunction,  and  at  one  of  its  nodes  at 
the  same  time,  in  order  that  a  transit  may  occur. 

Definition. — By  the  elongation  of  a  planet  from  the 
sun  is  meant  the  angle  subtended  at  the  earth  by  the  sun 
and  planet.  Thus  (fig.  33),  the  elongation  of  the  planet  P 
from  the  sun  is  the  angle  SEP. 

Corollary. — It  is  evident  that  the  elongation  of  an 
inferior  planet  must  always  be  an  acute  angle.  It  has  a 
maximum  value  at  a  point  near  P,  where  EP  becomes  a 
tangent  to  the  orbit  of  the  planet.  On  the  other  hand,  the 
elongation  of  a  superior  planet  may  have  any  value  from  0° 
to  180°,  reaching  the  latter  value  when  in  opposition. 

The  reader  should  bear  this  in  mind,  for  it  accounts  for 
the  fact  that  the  superior  planets  may  be  seen  at  all  angular 


CHAP.  VI.]  PHASES   OF   PLANETS.  81 

distances  from  the  sun,  occupying,  when  in  opposition,  a 
diametrically  opposite  point  on  the  celestial  'sphere,  so  as 
to  cross  the  meridian  at  midnight.  The  inferior  planets, 
Mercury  and  Venus,  on  the  other  hand,  on  account  of  their 
small  angular  distances  from  the  sun,  can  only  be  seen  either 
in  the  west  after  sunset  or  in  the  east  before  sunrise,  accord- 
ing to  their  position  relative  to  the  earth  and  sun. 

So  small  is  even  the  maximum  elongation  of  Mercury 
from  the  sun  that  it  is  only  possible  to  see  it  with  the  naked 
-eye  on  very  rare  occasions,  and  for  a  very  short  time  indeed 
after  sunset  or  before  sunrise. 


Phases  of  the  Planets. 

60.  As  the  planets  are  bodies  like  the  earth,  not  self- 
I'lminous,  but  deriving  their  light  from  the  sun,  they  can 
have  only  half  their  surface  lit  up  at  once,  the  other  half 
being  dark,  just  as  when  we  hold  a  globe  up  before  a  lamp, 
the   half    which   is   next   the    light   is   bright,   while  the 
half   which   is   turned   away    from    the   lamp    is   in    the 
shade.     Now  as  it  is  plain  that,  by  getting  into  different 
positions  with  respect  to  the  globe  and  the  lamp,  we  can  see 
as  much  or  as  little  as  we  choose  of  the  half  which  is  lighted 
up,  so  it  is  with  respect  to  the  planets  which  show  more  or 
less  of  their  illuminated  surface  to  us,  as  they  vary  their  posi- 
tions with  respect  to  the  earth  and  sun.     These  changes  in 
the  amount  which  we  see  of  the  illuminated  surface  of  a 
planet  are  called  its  phases.     It  is  needless  to  say  that  these 
phases  cannot  be  distinguished  with  the  naked  eye. 

61.  The  greatest  breadth  of  the  portion  of  the  illuminated 
surface  of  a  planet  which  is  turned  towards  the  earth  is  propor- 
tional to  the  exterior  angle  subtended  at  the  planet  by  the  earth 
and  sun. 

For  let  PS  and  PE  represent  the   directions  of  the 
sun  and  earth,  as  seen  from  P,  the  centre  of  the  planet. 

G 


A 


FIG.  34. 


82  THE   PLANETS.      SOLAR  SYSTEM.  [CHAP.  VI. 

Then  CD,  drawn  perpendicular  to  P89  separates  light  from 

darkness.      Erect  AB 

perpendicular  to  PE, 

then  the.  angle  APD 

will  measure  the   arc 

AD,    which     is    the 

greatest    breadth    of 

the  visible  illuminated 

surface,  as  seen  from 

the  earth.    The  angle 

8PE    is    the    angle 

subtended  by  the  sun  and  earth  at  P,  the  angle  SPF  being 

the  external  angle.     We  have  to  prove  the  L  APD  =  the 

LSPF. 

The  L  SPD  =  L  APF,  both  being  right. 

To  each  add  the  L  SPA  . 

.-.    the  L  APD  =  the  L  SPF,  .-.  &c. 

The  apparent  breadth  of  the  illuminated  surface  of  a  planet 
varies  as  the  versed  sine  of  the  exterior  angle  subtended  at  the 
planet  by  the  earth  and  sun. 

62.  We  have  already  seen  that  the  greatest  breadth  AD 
is  measured  by  the  exterior  angle  SPF.  But  the  apparent 
breadth  of  AD  will  be  measured  by  AM,  the  projection  of 
AD  on  a  line  perpendicular  to  PE,  the  direction  of  the 
observer.  For  the  earth  being  so  far  away  compared  with 
the  breadth  of  the  planet,  we  may  take  all  lines  drawn  from 
the  observer  to  the  surface  of  the  planet  as  being  parallel, 
and  each  perpendicular  to  AB. 

.'.     apparent  breadth  varies  as  AM. 
but  AM=  r  +PM  =  r  +  r  cos  BPD 


r  (1  -  cos  APD)  =  r  versin  APD 
=  r  versin  8PF. 


CHAP.  VI.] 


EXPLANATION    OF    PHASES. 


83 


Phases  of  Inferior  Planets. 

63.  Let  A  CDF  rep  resent  the  orbit  of  an  inferior  planet, 
E  being  the  earth,  and  8  the  sun.  We  will  now  suppose  the 
earth  to  be  at  rest,  and  that  the  planet  revolves  round  the 
sun  with  an  angular  velocity  equal  to  the  excess  of  its  real 
angular  velocity  over  that  of  the  earth  •  for  we  will  see  later 


0 


FIG.  35, 

on,  that  of  two  planets  the  nearer  to  the  sun  has  the  greater 
angular  velocity.  This,  of  course,  will  represent  exactly  the 
apparent  motion  of  the  planet,  as  seen  from  the  earth. 

When  the  planet  is  in  inferior  conjunction  at  A,  none  of 
its  illuminated  surface  is  seen. 

At  B  a  small  crescent  is  visible,  its  greatest  breadth 
being  measured  by  the  exterior  angle  SBD,  which  is  an 
acute  angle. 

G2 


84  THE    PLANETS.       SOLAR    SYSTEM.  [CHAP.  VI 

When  the  planet  reaches  its  greatest  elongation  from  the 
sun  at  C,  where  EC  is  a  tangent  to  its  orbit,  the  exterior 
angle  subtended  at  the  planet  is  a  right  angle,  and  the 
planet  appears  as  a  semicircle,  like  the  moon  at  first  or  third 
quarter.  It  is  then  said  to  be  dichotomized. 

At  D,  the  exterior  angle,  being  180°  -  SDE,  is  an  obtuse 
angle,  and  the  disc  appears  nearly,  but  not  quite,  full.  It  is 
then  said  to  be  gibbous. 

The  full  phase  is  reached  at  F,  and  the  above  changes  are 
then  repeated  in  inverse  order  until  inferior  conjunction  is 
again  reached. 

Superior  Planets9  Phases. 

64.  It  is  evident  that  a  superior  planet  must  always 
appear  either  full  or  gibbous ;  for  as  its  orbit  is  outside  that 
of  the  earth,  the  observer  is  always  on  the  same  side  of  the 
planet  as  the  sun  is  situated,  and  therefore  must  have  all,  or 
very  nearly  all,  its  illuminated  surface  turned  towards  him. 


FIG.  36. 

This  also  appears  from  the  fact  that  the  exterior  angle  sub- 
tended at  the  planet  is  always  obtuse,  and  therefore  the 
illuminated  disc  which  the  observer  sees  is  greater  than  a 
semicircle. 


CHAP.  VI.]  BRIGHTNESS  OF  PLANETS.  85 

It  is  easy  to  see  that  a  superior  planet  will  present  the 
smallest  portion  of  illuminated  surface  to  the  earth  when  the 
sun  and  planet  subtend  a  right  angle  at  the  earth  ;  that  is  to 
say,  a  superior  planet  is  most  gibbous  in  quadrature. 

For  let  Xloe  the  planet  (fig.  36).  Now  it  is  evident  that 
the  planet  will  appear  most  gibbous  when  the  exterior  angle 
at  X  is  least,  that  is,  when  the  angle  SXE  is  greatest,  which 
is  the  case  when  XE  is  a  tangent  to  the  orbit  of  the  earth 
(supposed  circular).  For  supposing  X  to  be  fixed,  and  that 
the  motion  is  due  to  the  earth  alone,  then  the  elongation  of 
the  earth  from  the  sun  as  seen  from  X,  viz.  the  angle  SXE9 
will  be  greatest  when  XE  becomes  a  tangent. 

Brightness  of  Planets. 

65.  The  brightness  of  a  planet  depends  on  the  amount  of 
illuminated  surface  turned  towards  the  earth  and  also  on  its 
distance.  For,  assuming  the  illuminated  surface  to  remain 
the  same,  it  will  appear  brighter  the  nearer  it  approaches  the 
earth,  the  intensity  of  the  brightness  being  inversely  as  the 
square  of  the  distance.  Thus,  at  twice  any  given  distance  it 
would  only  appear  one-quarter  as  bright ;  at  three  times  the 
distance  one-ninth  as  bright,  &c. 

The  inferior  planets  are  not  brightest  near  superior  con- 
junction, for  though  they  are  then  nearly  full  they  are  at 
their  greatest  distances  from  the  earth  :  Venus,  for  instance, 
when  at  superior  conjunction,  being  about  six  times  as  far 
away  as  when  in  inferior  conjunction  ;  its  disc  in  the  former 
case  subtending  an  angle  of  only  11",  and  in  the  latter  of  66". 
It  has  been  calculated  that  Venus  is  brightest  when  at  an 
elongation  of  about  40°  from  the  sun,  near  inferior  conjunc- 
tion, for  although  on  being  viewed  through  a  telescope  it 
then  only  appears  as  a  thin  crescent,  still,  owing  to  its 
proximity  to  the  earth,  that  crescent  appears  to  have  a  much 
larger  area  than  even  the  full  illuminated  circle  which  it 
presents  at  superior  conjunction. 


86 


THE   1'LANETS.       SOLAR  SYSTEM.  [CHAP.  VI. 


A  superior  planet  is  evidently  brightest  when  in  oppo- 
sition, for  not  only  does  it  then  appear  full,  but  it  is  at  the 
same  time  at  its  least  distance  from  the  earth. 


FIG.  37. 
Apparent  size  of  Venus  at  its  extreme  and  mean  distances  from  the  earth. 

To  find  the  ratio  of  the  distances  of  an  Inferior  Planet  and  the 
Earth  from  the  Sun  at  any  time. 

66.  Let  E,  V,  and  S  repre- 
sent the  earth,  Yenus,  and  the 
sun,  the  orbits  being  supposed 
circular  and  in  the  same  plane. 

The  L  SEV  is  got  byjob- 
servation,  and  the  L  JESVby 
calculation,  for  it  is  the  angle 
gained  by  Yenus  on  the  earth 
since  the  last  inferior  conjunc- 
tion, which  can  be  calculated 
as  follows : — 

Let  T  =  interval  between 
two  inferior  conjunctions,  in 
days,  and  x  =  number  of  days  since  last  inferior  conjunction  ; 


€HAP.  VI.]  PERIODIC  TIME  DETERMINED.  87 

.*.     Venus  gains  360°  on  earth  in  T  days  ; 

360° 
99        99     —  yT-        99        9>  1  day  ; 

360° 

99  99         -jT*X9>  99    X  fajZ'9 

.*.  the  L  ESV  is  known  and  all  the  angles  of  the  triangle 
SEV  are  known  ;  but  — 

SE     B 


o*.  the  ratio  of  SJE  to  8  Vis  found. 

Similarly  the  ratio  of  the  distances  of  a  superior  planet 
and  the  earth  from  the  sun  can  he  found,  the  proof  being  the 
same,  the  earth  in  this  case  gaming  on  the  planet. 

Definition.  —  The  periodic  time  of  a  planet  or,  as  it  is 
often  called,  its  sidereal  period,  is  the  time  taken  by  the  planet 
to  make  one  revolution  round  the  sun. 

The  synodic  period  is  the  interval  that  elapses  between 
two  conjunctions  of  the  same  kind  (both  inferior  or  superior) 
or  in  the  case  of  a  superior  planet,  between  two  oppositions. 

To  find  the  Periodic  Time  of  a  Planet  when  the  Synodic  Period 

is  known. 

INFERIOR  PLANET. 

67o  Let  P  =  periodic  time  of  planet  expressed  in  days, 
E=        „          „         earth          „  „ 

T  =  synodic  period  ; 

SfiO° 
v.  —  —  =  L  moved  through  by  planet  in  1  day, 

and        —  gr    =          „  „  earth 

360°     360°  ,  ,        ,  -     ,, 

c°.  -^  ---  —  =  angle  gained  by  planet  in  1  day,  for  the 
Jr          MI 

inferior  planet  goes  at  the  greater  rate  ; 


88  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.  VU 

60 
=7 


360° 
but         —=7-  =  angle  gained  by  planet  in  1  day  also  ; 


360°     360°     360 


•"     P 


--. 

XP     E     T  ' 

but  E  =  365-25  days,  and  T  being  known,  P  is  determined. 

EXAMPLE. 

The  interval  between  two  inferior  conjunctions  of  Mercury  is  116  days  :  find 
its  periodic  time. 

1_      I  I 

P     365-25  ~  116  ' 


nearly. 


Similarly  for  a  superior  planet  we  have  the  formula 

I_I  =  I 

E     P  "  T'  \ 

the  earth  going  at  the  greater  rate.  Thus  by  noting  the 
interval  between  two  inferior  conjunctions  (or  oppositions)  of 
a  planet  we  can  calculate  its  periodic  time,  supposing  its  orbit 
and  that  of  the  earth  circular. 

Kepler's  Three  Laws. 

68.  Kepler,  the  Danish  astronomer,  who  lived  at  the 
commencement  of  the  seventeenth  century,  first  enunciated 
the  following  laws  :  — 

i.  Each  planet  moves  in  an  elliptic  orbit  with  the  sun 
in  one  of  the  foci. 

ii.  The  straight  line  drawn  from  the  sun  to  a  planet 
(the  planet's  "  radius  vector  ")  sweeps  out  equal 
areas  in  equal  times. 

in.  The  squares  of  the  periodic  times  of  the  planets  are 
to  one  another  as  the  cubes  of  their  mean  distances 
from  the  sun. 


CHAP.  VI.] 


KEPLER  S    LAWS. 


Definition. — An  ellipse  is  a  plane  figure  bounded  by  one 
line  called  the  circumference, 
such  that  the  sum  of  the  dis- 
tances of  any  point  on  that 
circumference  from  two  fixed 
points  within  it  is  constant. 
Those  two  fixed  points  are 
called  the  foci. 

Thus  (see  fig.  39)  if  JPand 
Ff  be  the  two  foci  we  have 

EP+F'P  = 


FIG.  39. 

=  a  constant. 

The  following  is  therefore  a  mechanical  method  of  de- 
scribing an  ellipse : — 

Let  two  pins  be  taken  and  fixed  into  a  plane  board  or 
table,  say  at  F  and  F'  (fig.  39),  and  round  these  let  a  loose 
endless  string  be  thrown.  If  now  a  pencil  be  taken  and, 
keeping  the  string  tightly  stretched,  let  it  be  carried  round, 
occupying  successively  the  points  P,  Q,  R,  &c.,  the  curve 
traced  out  will  be  an  ellipse,  the  two  pins  JFandP'  being 
situated  at  the  foci. 

Kepler's  Second  Law  asserts  that  the  line  drawn  from  the 
sun  to  a  planet  sweeps  out 
equal  areas  in  equal  times, 
that  is,  if  the  times  of  de- 
scribing the  distances  AB 
and  PQ  are  equal,  then  the 
area  8AB  =  area  SPQ. 
From  this  we  can  conclude 
that  the  nearer  a  planet  ap- 
proaches the  sun  the  greater  must  be  its  velocity,  for  if  we 
regard  the  arcs  AB  and  PQ  as  being  described  in  a  small  unit 
of  time,  they,  being  small  compared  with  the  planet's  distance 
from  the  sun,  may  be  taken  as  straight  lines.  Now  if  the 
distance  from  S  to  AB  be  greater  than  from  S  to  PQ,  the 


FIG.  40. 


90  THE   PLANETS.      SOLAR   SYSTEM.  [CHAP.  VI. 

base  PQ  must  in  its  turn  be  greater  than  the  base  AB  in 
order  that  the  two  triangles  may  be  equal,  and  therefore  the 
velocity  at  PQ  is  greater  than  at  AB. 

Corollary. — As  the  earth  is  nearest  the  sun  at  mid- 
winter (Art.  46),  we  now  see  that  its  velocity  at  midwinter 
is  greater  than  at  any  other  part  of  its  orbit. 

Verification  of  Kepler' 's  Laws. 

69.  As  regards  the  earth  it  can  be  seen  by  measurement 
of  the  sun's  apparent  diameter  with  the  parallel  wire  or  other 
micrometer  that  the  orbit  is  not  a  true  circle,  and  that  there- 
fore the  earth's  distance  from  the  sun  is  not  constant,  being 
greatest  when  the  angle  subtended  by  the  sun's  diameter  is 
least.     We  can  now  therefore  construct  the   curve  which 
represents  the  earth's  orbit,  for  if  lines  be  drawn  from  a 
point  S  (fig.  40),  the  lengths  of  these  lines  being  inversely 
proportional  to  the  different  angles  which  the  sun's  diameter 
subtends  at  the  earth,  measurements  of  which  can  be  made 
daily,  the  extremities  of  these  lines  will  be  found  to  trace  out 
an  ellipse  with  S  in  one  of  the  foci. 

Kepler  determined  the  orbit  of  Mars  before  that  of  the 
earth  had  been  ascertained.  He  determined  the  position  of 
Mars  relative  to  the  earth  and  sun  by  a  method  somewhat 
similar  to  that  in  Art.  66,  and  by  this  means  he  arrived 
at  the  conclusion  that  the  orbit  of  Mars  was  elliptic.  The 
fact  that  he  considered  the  orbit  of  the  earth  as  circular  in 
these  calculations  did  not  give  rise  to  very  serious  error,  as 
the  eccentricity  of  the  earth's  orbit  is  very  small  and  much 
less  than  that  of  the  orbit  of  Mars. 

70.  Newton  showed  that  Kepler's  Third  Law  was  a  direct 
consequence  of  the  law  of  universal  gravitation  which  may  be 
enunciated  as  follows  : — 

Every  particle  in  the  universe  attracts  every  other  particle 
with  a  force  directly  proportional  to  the  mass  of  each,  and 
inversely  proportional  to  the  square  of  their  distance  apart. 


CHAP,  vi.]  BODE'S  LAW.  91 

To  deduce  Kepler's  Third  Law  from  the  Law  of 

Gravitation. 

Let  M  =  mass  of  the  sun. 

Let  r  and  r'  be  the  distances  of  two  planets  from  the  sun 
whose  periodic  times  are  Tand  Tf  respectively.  Now  by  the 
law  of  gravitation  the  attractions  of  the  sun  at  distances  r 
and  r  from  its  centre  are  in  the  proportion — 

M     M 

rZ       '    ^/2  ' 

But  the  centrifugal  acceleration  of  a  body  moving  in  a  circle 
of  radius  r  is  given  by  the  equation — 

t/  fJl%      ' 

therefore  assuming  the  orbits  of  the  planets  circular  we  have 

MM     4irzr     4irzr' 

^  '  V*  '' ''  ~Tr  ''   T/z  ' 

Multiplying  the  extremes  and  means  we  get  eventually — 

r'  r 

r*  f'l  ~    r'221Z  ' 

7^2 .  7^2  .  .  *,3  .  A/3 
•*•    •  -*•       •  •  *     •  i    , 

which  is  Kepler's  Third  Law. 

Bode's  Law. 

71.  There  is  a  remarkable  relation  between  the  distances 
of  the  different  planets  from  the  sun  which  bears  the  name 
of  the  astronomer  Bode.  Write  down  the  following  numbers 
in  which  each  after  the  first  is  doubled  : — 

0     1     2    4    8     16    32     64     128 
Now  multiply  by  3  and  add  4  to  each  and  we  get 

4    7     10     16    28     52     100     196     388 
corresponding  to  Mercury,  Venus,  Earth,  Mars,  Asteroids, 
Jupiter,  Saturn,  Uranus,  and  Neptune. 


92  THE   PLANETS.      SOLAR   SYSTEM.  [CHAP.  VI. 

These  numbers  are  approximately  proportional  to  the 
distances  of  the  different  planets  from  the  sun,  that  of  the 
earth  being  10.  There  is,  however,  a  serious  discrepancy  in 
the  case  of  Neptune,  which  is  represented  by  the  number  388, 
whereas  to  represent  its  actual  distance  it  should  300*369. 

This  law  received  a  remarkable  confirmation  from  the 
discovery  of  the  Asteroids,  which  consist  of  a  number  of 
small  planets  whose  orbits  lie  between  the  orbits  of  Mars  and 
Jupiter.  They  are  over  300  in  number.  Before  their  dis- 
covery there  was  no  planet  known  whose  distance  from  the 
sun  corresponded  to  the  number  28.  However  this  number 
is  found  to  approximately  represent  the  mean  distance  of  the 
different  asteroids  from  the  sun. 

Bode's  Law  can  be  expressed  by  means  of  the  general 
formula  D  =  4  +  3  x  2M~1,  where  D  represents  the  distance 
of  a  planet  from  the  sun,  and  n  the  number  of  the  planet 
beginning  with  Venus.  By  giving  to  n  the  values  1,  2, 
3,  &c.,  the  numbers  corresponding  to  the  distances  of  the 
different  planets  from  the  sun  commencing  with  Venus, 
are  found  to  be  the  same  as  those  mentioned  above. 


True  Distance. 

Mercury, 

r     ,<  \,                  4 

= 

4 

3-871. 

Venus, 

,tv      4  +  3x2° 

= 

7 

7'233. 

Earth, 

4  +  3X21 

= 

10 

10-000. 

Mars, 

.  '     4  +  3  x  22 

= 

16 

15-237. 

Asteroids, 

4  -1-  3  x  23 

= 

28 

22  to  31. 

Jupiter, 

4  +  3x2* 

= 

52 

52-028. 

Saturn, 

4  +  3  x  25 

= 

100 

95-388. 

Uranus, 

.        4  +  3  x  26 

= 

196 

191-826. 

Neptune, 

.        4+3x2* 

= 

388 

300-369. 

Direct  and  Retrograde  Motion.  Stationary  Points. 
72.  A  planet's  apparent  motion  is  said  to  be  direct  when 
it  seems  to  move  in  the  same  direction  as  the  sun  in  the 
ecliptic,  and  retrograde  when  it  appears  to  move  in  a  contrary 
direction.  In  other  words,  its  motion  is  direct  when  its 
longitude  is  increasing,  and  retrograde  when  diminishing. 


CHAP.   VI.]  DIRECT  AND  RETROGRADE  MOTION.  93 

As  the  earth  E  moves  in  its  orbit  in  the  direction  indi- 
cated by  the  arrow  (fig.  41)  the  sun  appears  to  move  as  if  the 
line  E8  were  rotating  round  E  from  right  to  left  in  a  direction 
contrary  to  the  hands  of  a  watch.  Therefore  we  will,  for 
greater  clearness,  regard  a  contra- watch-hand  rotation  round 
E  as  direct,  and  a  watch-hand  rotation  as \  retrograde,  E  being 
supposed  fixed. 

An  inferior  planet  moves  with  greater  velocity  than 
the  earth  (Art.  47)  ;  therefore  when  the  planet  is  at  inferior 
conjunction  the  extremity  V  of  the  line  VE  moving  with 
greater  velocity  than  E,  the  line  will  appear  to  revolve  round 


FIG.  41. 

E  like  the  hands  of  a  watch,  and  therefore  the  apparent  motion 
of  a  planet  at  inferior  conjunction  is  retrograde.  At  the  points 
of  greatest  elongation  P  and  Q  the  planet's  own  velocity 
will  produce  no  change  in  its  direction,  as  it  is  along  the 
lines  EP  and  EQ,  respectively,  but  the  earth's  motion  will 
make  the  lines  EP  and  EQ,  appear  to  revolve  in  a  direction 
contrary  to  that  of  the  hands  of  a  watch.  Hence  the  planet's 
apparent  motion  at  P  and  Q  is  direct.  At  any  point  on  the 


94  THE  PLANETS.      SOLAR  SYSTEM.  [CHAP.  VI. 

arc  P  V'Q,  the  motion  will  appear  direct,  for  both  the  planet's 
own  velocity  and  that  of  the  earth  will  combine  to  make  the 
line  joining  them  revolve  round  E  in  a  direction  contrary  to 
that  of  the  hands  of  a  watch. 

Again,  as  the  planet's  motion  appears  retrograde  at  Vr 
and  direct  at  P  and  Q,  it  must  pass  through  two  points  m 
and  n,  at  which  the  retrograde  motion  is  on  the  point  of 
changing  into  direct  or  vice  versa,  and  at  which  the  planet 
does  not  seem  to  move.  These  two  positions  are  called  the 
stationary  points. 

A  superior  planet  on  the  other  hand  moves  with  a  velo- 
city which  is  less  than  that  of  the  earth ;  therefore  when 
the  planet  is  in  opposition  at  the  point  M  (fig.  41)  the 
line  EM  will  appear  to  rotate  round  E  like  the  hands  of  a 
watch.  Hence  the  motion  of  a  superior  planet  in  opposition  is 
retrograde. 

Again,  when  the  planet  is  in  quadrature  at  X  and  Y  the 
velocity  of  the  earth  will  have  no  effect,  as  it  is  in  the  direction 
of  the  line  joining  the  observer  to  the  planet.  The  planet's 
own  motion,  however,  will  cause  EX  or  JEYto  revolve  round 
E  in  a  contra- watch-hand  direction.  Hence  the  apparent 
motion  of  a  superior  planet  in  quadrature  is  direct. 

Also  at  any  position  along  the  arc  JOf'P'both  the  planet's 
velocity  and  that  of  the  earth  combine  to  cause  the  line  join- 
ing them  to  appear  to  revolve  with  a  contra-watch-hand 
rotation,  i.e.  the  planet's  motion  will  be  direct. 

As  the  planet  appears  retrograde  at  M  and  direct  at  JTand 
P,  there  will  be  two  points  p  and  q  at  which  the  retrograde 
motion  is  on  the  point  of  changing  into  the  direct,  and  vice 
versa,  these  points  being  the  two  stationary  points  of  the 
planet. 

Rotations  of  the  Planets  round  their  Axes. 

73.  We  have  already  seen  that  the  earth  and  sun  rotate. 
It  has  also  been  shown  by  observing  the  markings  and 


CHAP.  VI.]         ROTATIONS  OF  PLANETS.  95 

on  the  surfaces  of  the  planets  that  most  of  them,  and  probably 
all,  rotate  in  the  same  manner.  Mars  rotates  once  in  24h  37m, 
so  that  a  day  in  Mars  is  almost  of  the  same  length  as  a  day 
on  the  earth.  Jupiter  takes  9h  55m,  and  Saturn  10h  29m. 

It  is  much  more  difficult  to  find  the  period  of  rotation  of 
an  inferior  than  of  a  superior  planet ;  for  the  latter  when  in 
opposition  can  be  seen  all  night,  whereas  an  inferior  planet 
only  appears  as  an  evening  or  morning  star,  and  observations 
can  only  be  made  at  intervals  of  24  hours.  Now  supposing 
that  markings  are  observed  on  the  surface  of  Venus  after 
sunset,  it  is  found  that  they  occupy  nearly  the  same  position 
on  the  following  night.  From  this  we  might  be  led  to  one 
or  other  of  two  conclusions — (1)  either  Yenus  makes  a  revo- 
lution on  its  axis  in  about  24  hours,  or  (2)  it  takes  a  very 
long  period  to  complete  a  revolution  so  that  the  angle  turned 
through  in  24  hours  would  be  very  small.  In  either  case  it 
is  evident  the  markings  would  not  be  much  changed  during 
24  hours.  Until  quite  recently  it  was  believed  that  the  first 
conclusion  was  true.  From  observations  by  Schrater  the 
period  for  Yenus  was  believed  to  be  23h  21m,  and  for  Mercury 
24h  5m.  Professor  Schiaparelli,  however,  has  recently  con- 
tended that  Mercury  and  Yenus  take  the  same  time  to  rotate 
on  their  axes  as  they  do  to  revolve  round  the  sun,  the  period 
for  the  former  being  88  and  the  latter  224  days,  and  that 
therefore  they  turn  always  nearly  the  same  face  towards  the 
sun,  just  as  the  moon  does  to  the  earth,  large  portions  of  each 
being  in  perpetual  sunlight,  and  other  portions  always  in 
darkness. 

As  these  planets  are  only  to  be  seen  close  to  the  horizon 
after  sunset  or  before  sunrise  the  changes  in  the  temperature 
and  density  of  the  lower  strata  of  the  atmosphere  render  it 
very  difficult  to  observe  the  markings  on  their  surface  with 
sufficient  accuracy  to  determine  the  exact  truth ;  but  more 
recent  observations  would  seem  to  show  that  Schiaparelli's 
conclusion  is,  at  all  events,  false  in  the  case  of  Yenus. 


96  THE    PLANETS.       SOLAR    SYSTEM.  [CHAP.  VI. 

To  prove  that  the  Velocities  of  Two  Planets  round  the  Sun  are 
inversely  as  the  square  roots  of  their  distances  from  the  Sun. 

74.  For  by  Kepler's  Third  Law  we  have 

Tz:  r2::r3:/3; 

but  circumference  of  orbit  =  velocity  x  time. 
.-.    2wr  =  vT-, 


.,  /2wrV  /27T/V 

therefore  we  have  (  -  j  :  (  —  —  J  :  :  rz  :  rz  ; 


r' 


.•.     vi  v'  :  : 

Corollary.  —  Hence  of  two  planets  the  nearer  to  the  sun 
has  the  greater  velocity. 

75.  We  shall  conclude  this  chapter  with  a  brief  review  of 
the  different  bodies  which  constitute  the  solar  system. 

Mercury  ?  . 

This  is  the  nearest  planet  to  the  sun.  Its  diameter  is  about 
3000  miles,  being  much  smaller  than  that  of  the  earth  (e), 
whose  diameter  is  8000  miles.  The  orbit  of  Mercury  is  much 
more  eccentric  than  those  of  the  other  principal  planets,  that 
is,  it  does  not  so  nearly  approach  a  circular  shape.  At  one 
time  the  planet  approaches  to  within  28,000,000  miles  of  the 
sun,  and  again  in  the  opposite  point  of  its  orbit  recedes  to  a 
distance  of  43,000,000  miles.  It  is  also  distinguished  by  the 
great  inclination  of  its  orbit  to  the  ecliptic,  namely,  about  7°. 
Its  periodic  time  about  the  sun  is  about  88  days. 


CHAP.  VI.]  TRANSITS  OF  VENUS.  97 

Venus    $  . 

Venus  has  a  diameter  almost  equal  to  that  of  the  earth. 
Its  orhit  also  like  that  of  the  earth  differs  but  little  from  a 
circle.  The  inclination  of  its  orbit  to  the  ecliptic  is  3°  23'. 
AVe  have  seen  how  Mercury  and  Venus  being  inferior  planets 
ure  only  to  be  seen  within  certain  angular  distances  on  the 
east  or  west  side  of  the  sun,  and  are  therefore  morning  or 
evening  stars ;  and  also  that  their  discs,  when  seen  through 
a  telescope,  show  phases  like  those  of  the  moon.  In  both 
these  planets  it  has  been  observed  that  the  line  of  separation 
of  the  light  from  the  dark  portions  is  not  continuous  but 
notched,  and  also  that  the  horns  of  the  crescents  they  present 
are  sometimes  cut  off  abruptly.  This  is  caused  by  mountains 
on  their  surfaces,  which  have  been  calculated  to  rise  in  both 
planets  to  heights  considerably  greater  than  those  on  the 
earth.  The  periodic  time  of  Venus  is  ^24 -7  days. 

Transits  of  Venus  and  Mercury. 

76.  We  have  already  seen  that  the  transit  of  Venus  or 
Mercury  can  only  occur  when  the  planet  is  in  inferior  con- 
junction at  or  near  one  of  its  nodes. 

If  a  transit  of  one  of  these  planets  occur  at  any  time 
another  transit  at  the  same  node  will  not  occur  until  the  earth 
and  the  planet  shall  have  each  made  an  exact  number  of 
revolutions. 

Now  8  revolutions  of  the  earth  expressed  in  days  are 
almost  equal  to  an  exact  number  of  revolutions  of  Venus,. 
viz.  13,  there  being  only  a  difference  of  one  day,  for 

8  x  365-242  =  2922  days  nearly, 
and 

13  x  224-7      -  2921-1  days. 

Hence,  if  a  transit  of  Venus  occur  at  any  time  there  may. 
be  another  at  the  same  node  8  years  afterwards  if  one  has  not 

u 


98  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.  VI. 

already  occurred  8  years  before.  There  will  not,  however, 
be  a  transit  16  years  afterwards,  as,  on  account  of  the  above 
difference  of  one  day,  the  distance  from  the  node  when  in 
conjunction  will  be  too  great  In  fact,  a  transit  at  the  same 
node  cannot  in  this  case  occur  for  another  235  years,  whicli 
is  the  next  number  of  years  which  corresponds  to  an  exact 
number  of  revolutions  of  Yenus,  for 

235  x  365-242  =  85835  days  nearly, 
and 

382x2247      =85835     „        „ 

The  first  transit  of  Venus  ever  observed  was  that  seen  by 
Horrox,  in  1639,  which  occurred  at  the  ascending  node.  A 
transit  at  this  node  did  not  again  occur  for  235  years,  viz.  in 
1874,  and  again  in  1882.  Transits  at  the  descending  node 
have  been  observed  in  the  years  1761  and  1769,  the  next 
occurring  in  the  year  2004. 

Transits  of  Mercury  occur  more  frequently  than  those  of 
Yenus,  for  its  periodic  time  is  such  that  it  more  frequently 
happens  that  an  exact  number  of  revolutions  of  the  planet 
correspond  to  an  exact  number  of  years.  Thus  transits  of 
Mercury  at  the  same  node  may  happen  at  intervals  of  7,  13, 
33,  or  46  years. 

At  present  the  earth  in  its  orbital  motion  is  opposite  a 
node  of  Yenus  on  the  5th  of  June,  and  again  on  the  7th  of 
December.  Hence,  for  a  very  long  period  of  time,  transits 
of  Yenus  will  occur  in  December  and  June.  For  a  similar 
reason  transits  of  Mercury  will  occur  in  May  and  November. 

Transits  of  Yenus  are  of  great  practical  interest,  as  their 
observation  furnishes  the  most  accurate  methods  of  deter- 
mining the  sun's  parallax  and  distance  (Chapter  VII.). 
Transits  of  Mercury  cannot  be  used  in  the  same  manner,  as 
its  distance  from  the  earth  approaches  too  nearly  that  of  the 
sun  to  give  reliable  results.  Besides  it  moves  too  rapidly 
across  the  sun's  disc  to  give  time  for  accurate  observations ; 


•CHAP.  VI.]  MARS.  99 

?and  also  as  its  orbit  does  not  so  nearly  approach  a  circular 
shape  as  that  of  Venus,  the  ratio  of  its  distance  from  the 
sun  to  that  of  the  earth  cannot  be  so  easily  calculated. 

Mars  $ . 

77.  Tho  nearest  of  the  superior  planets  is  Mars.  Its 
-distance  from  the  sun  varies  from  127,000,000  to  153,000,000 
miles,  and  therefore  its  orbit  is  much  more  eccentric  than 
that  of  the  earth.  If  the  orbits  of  Mars  and  the  earth  were 
both  circular  the  planet  would  be  closest  to  us  at  opposition, 
its  distance  being  then  only  the  difference  of  the  radii  of  the 
orbits.  But  the  distance  of  Mars  from  the  sun,  as  we  have 
seen  above,  is  very  variable,  and  that  of  the  earth  from  the 
sun  changesfrom  90,500,000  miles  at  midwinter  to  93,500,000 
miles  at  midsummer.  We  can,  therefore,  see  how  some 
oppositions  are  much  more  favourable  for  observation  than 
•others.  For,  suppose  during  opposition  that  Mars  were  at 
its  least  distance  from  the  sun,  and  the  earth  at  its  greatest 
distance,  the  planet  would  only  be  distant  from  us  by 
34,000,000  miles,  and  astronomers  would  then  have  the 
opportunity  of  viewing  it  under  most  favourable  conditions. 
If  we  imagine  the  two  points  where  Mars  is  nearest  to 
and  at  its  greatest  distance  from  the  sun  to  be  joined  to  one 
another,  it  is  found  that  the  earth  in  its  orbital  motion  passes 
close  to  this  line  on  August  26,  and  again  on  February  22. 
On  August  26  the  earth  passes  that  point  of  its  orbit  which 
j  is  in  a  line  between  the  sun  and  the  position  which  Mars 
I  would  occupy  when  closest  to  the  sun  (perihelion] ;  and  on 
February  22  it  crosses  the  line  between  the  sun  and  the 
point  in  which  Mars  would  be  situated  when  at  its  greatest 
distance  from  the  sun  (aphelion).  Therefore  if  we  regard  the 
orbit  of  the  earth  as  being  circular  (for  it  is  much  more 
nearly  so  than  that  of  Mars),  the  nearer  the  date  of  an  opposi- 
tion approaches  to  August  26,  the  more  favourable  are  the 
conditions  under  which  the  planet  can  be  observed,  and  the 

H2 


100  THE   PLANETS.      SOLAR   SYSTEM.  [CHAP.  VI- 

closer  that  date  is  to  February  22  the  more  unfavourable  i» 
such  an  opposition  for  accurate  observation. 

The  periodic  time  of  Mars  is  about  687  days,  and  the- 
inclination  of  its  orbit  to  the  ecliptic  is  about  2°.  Two  small 
satellites  of  Mars  were  discovered  by  Mr.  Hall,  of  Washington, 
during  the  opposition  which  occurred  on  epteinber  5,  1877, 
when  the  planet  was  very  close  to  the  earth,  the  date  of  tlie 
discovery  being  only  ten  days  after  the  best  date  possible. 
They  have  been  named  Deimos  and  Phobos,*  the  former, 
which  is  the  outer,  completing  a  revolution  round  Mars  in 
about  30h  18m,  and  the  latter  in  7h  39m.  As  Mars  completes 
a  revolution  on  its  axis  in  24h  37m,  we  have  in  Phobos  an 
example  of  a  satellite  revolving  round  the  primary  planet 
much  more  quickly  than  the  latter  rotates  on  its  axis,  a  case 
which  is  without  a  parallel  in  the  solar  system. 

At  the  beginning  of  this  century  a  number  of  very  small 
planets  were  discovered  with  orbits  lying  between  Mars  and 
Jupiter.  They  are  called  the  Asteroids.  There  are  at 
present  considerably  over  300,  the  smaller  ones  being  only 
a  few  miles  in  diameter.  The  four  largest  are  Vesta,  Juno, 
Ceres,  and  Pallas.  Their  orbits  are  generally  very  eccentric, 
some  of  them  being  also  inclined  at  considerable  angles  to  the 

ecliptic. 

Jupiter  if. 

78.  This  is  the  largest  of  all  the  planets,  its  diameter 
being  11  times  that  of  the  earth.  Its  orbit  is  nearly  circular 
like  that  of  the  earth,  and  is  inclined  to  the  ecliptic  at  an 
angle  of  about  1-^°.  When  observed  through  a  telescope,  a 
number  of  bright  belts  or  bands  are  seen  encircling  it  parallel 
to  its  equator,  which  are  probably  belts  of  clouds  or  vapours 

*  In  Homer  Deimos  and  Phobos  are  represented  as  the  attendants  of  Mars. 
The  passage  in  the  Iliad  which  first  suggested  the  names  of  the  satellites  he 
been  thus  construed  by  Professor  Tyrrell : — 

"  Mars  spa"ke  and  called  Dismay  and  Rout 
To  yoke  his  steeds,  and  he  did  on  his  harness  sheen." 

II.  15,  119,  120. 


CHAP.  Vic]  SATURN.  101 

in  its  atmosphere.  It  has  five  satellites  or  moons,  which  may 
he  seen  with  a  good  field  glass.  The  periodic  times  of  these 
five  moons,  and  their  mean  distances  from  Jupiter,  satisfy 
Kepler's  third  law,  which  we  have  seen  is  true  for  the  orbits 
of  the  planets  round  the  sun.  This  is  true  of  the  satellites 
of  all  the  planets.  They  are  frequently  eclipsed  when  they 
enter  the  shadow  cast  hy  Jupiter  on  the  side  opposite  the 
sun.  An  eclipse  must  not,  however,  be  confounded  with  an 
occultation  which  happens  when  a  satellite  is  in  a  line  with 
Jupiter  and  the  earth  so  as  to  be  hidden  from  the  observer's 
view.  Again,  a  very  curious  phenomenon  is  observed  when 
a  satellite  comes  between  Jupiter  and  the  sun.  The  shadow 
•cast  by  the  satellite  will  then  be  observed  as  a  dark  spot 
moving  across  the  face  of  Jupiter,  which  is  indeed  a  wonder- 
ful sight,  illustrating,  as  it  does,  the  appearance  which  the 
«arth  would  present  if  viewed  from  Mercury  or  Venus  during 
a  total  eclipse  of  the  sun  by  our  satellite  the  moon.  A  transit 
of  the  satellite  may  also  occur  when  the  satellite  is  in  a  direct 
line  between  Jupiter  and  the  earth. 

Saturn  J? . 

79.  The  orbit  of  Saturn  is  also  very  nearly  circular.  It  is 
inclined  to  the  ecliptic  at  an  angle  of  about  2  j°.  At  certain 
periods  Saturn  when  viewed  through  a  telescope  presents  a 
most  wonderful  appearance.  It  is  surrounded  by  a  series  of 
circular  rings  which  do  not  touch  the  surface  of  the  planet ; 
indeed  through  the  interval  between  the  rings  and  the  body 
of  the  planet  fixed  stars  are  sometimes  seen.  The  plane  of 
these  rings  is  inclined  at  a  constant  angle  of  about  28°  to 
the  plane  of  Saturn's  orbit,  and,  therefore,  they  being  seen 
obliquely  by  us,  will  not  appear  circular  but  oval.  They  are 
supposed  to  be  formed  of  immense  numbers  of  small  satellites. 
They  become  invisible — (1)  when  the  plane  of  the  rings, 
when  produced,  passes  through  the  earth,  for  being  very  thin 
when  the  edge  is  turned  towards  us,  it  is  not  possible  to  see 


102  THE   PLANETS.      SOLAR   SYSTEM.  [CHAP.  VJ. 

them  ;e&0ept  through  the  most  powerful  telescopes ;  (2)  when 
their  plane  passes  through  the  sun,  for  they,  deriving  their 
light  from  the  sun,  have  only  their  edge  illuminated; 
(3)  when  their  plane  passes  between  the  earth  and  the  sun, 
for  their  dark  surface  being  towards  us  it  is  not  possible  to* 
see  them., 

Saturn  has,  besides,  eight  satellites,  all  situated  external 
to  the  rings.  The  seven  nearest  move  in  orbits  whose  planes 
almost  coincide  with  the  plane  of  the  rings,  but  that  of  the 
eighth  is  inclined  to  this  plane  at  an  angle  of  about  10°. 


7 


FIG.  42. — Phases  of  Saturn's  Rings. 

Uranus  y. 

Uranus  was  discovered  in  1781  by  Herschel.  Its  orbit  is 
nearly  circular,  and  inclined  at  a  very  small  angle  to  the 
ecliptic.  Four  satellites  have  been  discovered  which  revolve 
in  orbits  nearly  perpendicular  to  the  plane  of  the  orbit  of 
Uranus. 

Neptune  ^ , 

The  discovery  of  Neptune  is  one  of  the  most  brilliant  in 
the  history  of  Astronomy.  It  was  found  that  the  positions 
which  it  was  calculated  Uranus  should  occupy,  after  making 
allowance  for  all  known  disturbing  forces,  did  not  coincide 
with  the  observed  positions.  It  was  therefore  thought  that 
there  must  be  some  unknown  planet  whose  attraction  produced 
these  disturbances.  After  the  most  laborious  calculations  the 


CHAP.  VI.]  COMETS.  103 

position  which  this  unknown  body  should  occupy  was  deter- 
mined at  almost  the  same  time  by  Leverrier  in  France,  and 
Adams  in  England,  in  the  year  1846.  One  satellite  of 
Neptune  has  been  discovered. 

Comets. 

80.  The  solar  system  includes  a  number  of  other  bodies 
which  differ  widely  from  the  planets,  both  in  their  physicaj 
state  and  in  the  nature  of  the  orbits  described  by  them  round 
the  sun.  These  bodies  are  called  comets.  Comets  differ  very 
much  as  regards  their  shape,  and  even  the  shape  and  size  of 
the  same  comet  may  change  considerably  at  different  parts  of 
its  orbit ;  but  we  generally  find  at  one  end  a  brilliant  nucleus 
surrounded  by  nebulous  matter  stretching  out  into  an  elon- 
gated tail.  The  tails  of  some  comets  which  have  appeared 
have  been  of  enormous  dimensions;  that  of  1811  had  a  tail 
23°  in  length,  another  in  1843  had  a  length  of  40°,  while 
that  of  1618  extended  across  the  sky  through  an  arc  of  104°. 

Comets  generally  appear  suddenly  in  the  sky,  remaining 
visible  for  some  weeks,  or  months,  during  which  time  they 
approach  the  sun  with  great  velocity ;  they  then  recede  from 
it,  and  finally  disappear  from  view. 

The  mass  and  density  of  comets  are  extremely  small ;  it  is 
even  possible  to  see  faint  stars  shining  through  them  almost 
as  if  no  material  body  were  interposed  between. 

By  far  the  greater  number  of  comets  describe  orbits  of 
such  great  eccentricity  that  we  may  regard  them  as  parabolas 
described  round  the  sun  as  focus,  the  other  focus  being 
practically  at  an  infinite  distance.  But  there  are  a  few 
comets  whose  orbits,  although  much  more  elongated  than 
those  of  the  planets,  are  sufficiently  small  to  be  contained 
within  the  solar  system.  The  motions  of  these  can  be  cal- 
culated and  the  dates  of  their  return  predicted  from  knowing 
1  he  magnitudes  of  the  ellipses  which  they  describe.  These 
.ire  called  periodic  comets. 


104  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.  VI. 

The  orbits  of  comets,  besides  being  much  more  eccentric, 
also  differ  from  those  of  the  planets  in  that  they  may  be 
inclined  at  any  angle  to  the  plane  of  the  ecliptic  ;  moreover, 
all  the  planets  go  round  the  sun  in  the  same  direction  as 
the  earth  moves,  whereas  the  motion  of  some  comets  is  direct 
and  of  others  retrograde. 

Periodic  Comets. 

81.  H  alley's  Comet. — Of  the  periodic  comets,  that 
known  as  Halley's  is,  perhaps,  the  most  remarkable.  It  was 
observed  that  the  comets  which  appeared  in  the  years  1531, 
1607,  and  1682  were  almost  identical  as  regards  the  position 
of  the  nodes,  the  perihelion  distance  from  the  sun,  the  incli- 
nation of  the  orbit  to  the  plane  of  the  ecliptic,  and  certain 
other  measurements,  from  which  Halley  concluded  that  they 
were  really  one  and  the  same  comet,  having  a  periodic  time 
about  the  sun  of  75  years,  and  he  therefore  predicted  its 
return  in  1758.  Clairaut,  having  calculated  that,  owing  to 
perturbations  caused  by  the  attractions  of  Jupiter  and  Saturn, 
it  would  be  retarded  518  days  and  100  days  respectively, 
predicted  that  it  would  be  closest  to  the  sun,  or,  in  other 
words,  at  perihelion,  at  about  the  middle  of  April,  1759. 
No  allowance  was  made  for  disturbances  caused  by  the 
attractions  of  Uranus  and  Neptune,  as  these  planets  were 
not  then  discovered.  It  actually  appeared  at  the  end  of 
1758,  and  reached  the  perihelion  at  the  middle  of  March, 
1759. 

Halley's  Comet  has  since  appeared  in  1835,  and  it  may 
be  again  expected  in  1910.  One  of  the  previous  visits  of 
Halley's  Comet  was  on  a  very  memorable  occasion  in  the  year 
1066,  the  date  of  the  Norman  conquest ;  the  picture  of  this 
comet  is  depicted  in  the  Bayeux  tapestry. 

Encke's  Comet. — This  periodic  comet  is  also  known  to 
describe  an  elliptic  path  round  the  sun.  At  perihelion  it  is 
closer  to  the  sun  than  Mercury  ;  and  at  aphelion,  at  its 


CHAP.  VI.]  COMETS.  105 

greatest  distance,  it  is  not  altogether  as  far  from  the  sun  as 
Jupiter;  so  that  its  orbit  is  well  within  the  limits  of  the 
solar  system.  Its  periodic  time  is  about  3|  years.  The 
motion  of  this  comet  has  been  most  carefully  observed,  and 
the  perturbations  in  its  movements  due  to  the  attractions  of 
the  earth  and  the  other  planets  have  been  calculated.  But 
it  is  found  that  after  making  allowance  for  all  these  disturb- 
ing forces,  there  is  still  a  diminution  in  its  periodic  time  of 


FIG.  43. — Orbits  of  some  Periodic  Comets. 

2J  hours  in  each  successive  revolution.  Encke  accounted  for 
this  by  supposing  that  there  exists  for  a  considerable  distance 
round  the  sun  a  medium  which,  although  of  extreme  tenuity, 
is  still  capable  of  offering  sufficient  resistance  to  the  passage 
of  a  body  of  such  small  density  as  a  comet  as  to  appreciably 
diminish  its  periodic  time. 

Non-periodic  comets  are  much  more  numerous  than  peri- 
odic. To  this  class  belonged  the  great  comet  of  1843,  Donati's 
(jomet,  which  appeared  in  1858,  and  the  comet  of  1881. 


106  THE    PLANETS.       SOLAR   SYSTEM.  [eHAP.  VI, 

Meteors  or  Shooting  Stars. 

82.  In  addition  to  the  different  members  of  the  solar 
system  which  we  have  already  enumerated  there  are  an 
innumerable  number  of  minute  bodies  which  are  called 
meteors.  When  these  bodies,  which  move  with  great 
velocity,  impinge  on  the  earth's  atmosphere  in  a  direction 
opposite  to  that  in  which  the  earth  is  moving,  the  relative 
velocity  is  so  large  that  the  heat  developed  by  the  resistance 
of  the  air  is  sufficient  to  consume  them,  and  they  appear  as  a 
streak  of  light  in  the  sky.  Others,  whose  relative  velocity  is 
not  so  great,  on  rare  occasions,  fall  to  the  earth  unconsumed. 
These  are  called  meteorites  or  meteoric  stones.  The  heights 
of  meteors  have  been  found  to  vary  from  16  to  160  miles. 

Although  it  is  possible  to  see  many  stray  meteors  on 
almost  any  night,  there  are  three  periods  in  the  year  at  which 
they  occur  in  very  considerable  numbers,  viz.  August  9-11, 
November  12-14,  and  November  27-29. 

Radiant  Point. — During  these  August  and  November 
meteoric  showers  the  apparent  paths  on  the  celestial  sphere  of 
most  of  the  meteors  seem  to  spring  from  one  common  point 
called  a  radiant  point.  This  is  merely  an  effect  of  perspective. 
For,  as  we  will  shortly  see,  all  the  meteors  which  compose 
the  swarm  through  which  the  earth  happens  to  be  passing 
are,  for  the  short  period  they  are  under  observation,  approxi- 
mately moving  in  parallel  straight  lines.  If  now  we  imagine 
planes  drawn  through  these  lines  and  the  observer  they  will 
cut  the  celestial  sphere  in  a  number  of  great  circles  all  having 
a  common  diameter,  viz.  the  line  drawn  through  the  observer 
parallel  to  the  common  direction  of  the  motion  of  the  meteors. 
This  line  when  produced  will  cut  the  celestial  sphere  in  the 
two  common  points  of  intersection  of  all  the  circles,  one 
of  which  is  the  radiant  point.  The  radiant  point  for  the 
August  meteors  is  in  the  constellation  of  Perseus,  and  those 
for  the  two  showers  which  take  place  in  November  are  in 


.CHAP.  VI. j          COMETS   AND   METEORS.       APSIDES.  107 

the  constellations  of  Leo  and  Andromeda  respectively.   Hence 
we  have  the  three  showers — 

The  Perseids,        Aug.    9-11,  radiant  point  in  Perseus. 
The  Leonids,         Nov.  12-14,  „          in  Leo. 

The  Andromedes,  Nov.  27-29,  „      in  Andromeda. 

Connexion  between  Comets  and  Meteors. 

83.  The  meteors  which  we  see  dashing  into  the  atmo- 
sphere  at   about  the  14th  November   are   believed  to    t>e 
portions  of  a  train  of  an  innumerable  number  of  minute 
bodies  whose  orbits  are  almost  identical  with  that  of  Temple's 
Comet.     The  orbit  of  this  comet  actually  cuts  that  of  the 
earth,  the  earth  arriving  each  year  at  the  point  of  intersection 
on  the  14th  November.     There  are  portions  of  this  elliptic 
belt  where  these  minute  bodies  are  crowded  together  into 
groups;  and  on  certain  occasions,  separated  by  long  intervals 
of  time,  the  earth  passes  through  one  of  these  groups  or 
shoals,  as,  for  instance,  on  November  13,  1866,  when  the 
appearance  of  the  heavens,  lit  up  by  myriads  of  meteors,  was 
of  the  most  wonderful  description. 

As  the  periodic  time  of  this  shoal  is  33  years,  it  was  fully 
expected  that  an  equally  brilliant  display  would  be  observed 
in  November,  1899.  But  although  a  few  individual  members 
of  the  band  were  observed  from  several  places  on  the  earth, 
the  result,  for  reasons  which  we  can  at  present  only  con- 
jecture, was  very  disappointing. 

The  showers  known  as  the  Andromedes  and  Perseids  can 
be  similarly  accounted  for,  the  former  being  due  to  the  fact 
that  the  orbit  of  Biela's  Comet  cuts  that  of  the  earth  at  a 
point  corresponding  to  November  27. 

84.  Line  of  Apsides.— Those  points  where  the  earth,  or 
a  planet,  in  its  orbit  is  nearest  to  and  most  remote  from  the 
sun  (perihelion  and  aphelion)  are  called  apsides.     The  earth 


108  THE    PLANETS.       SOLAR   SYSTEM.  [CHAP.  YI. 

is  nearest  the  sun  at  the  apse  A  (fig.  44)  on  the  31st  December, 
and  arrives  at  the  opposite  apse  B  on  the  1st  July,  the  line 
AB  being  called  the  line  of  Apsides.  This  line  evidently 
coincides  with  the  major  axis  of  the  ellipse. 

N.B. — The  sun's  apparent  diameter  is  greatest  when 
the  earth  is  at  the  apse  A,  and  least  when  at  B.  Also 
as  observed  from  two  positions  E  and  Ef  such  that  the 
L  ASE  =  L  ASE'  the  sun  will  have  the  same  apparent 
diameter,  for  from  the  symmetry  of  the  ellipse  it  is  evident 
that  the  distances  SE  and  81?  are  equal. 


FIG.  44. 

To  find  the  Direction  of  the  Apse  Line. — It  might 

at  first  suggest  itself  that  the  direction  of  the  line  of  Apsides 
could  easily  be  found  by  noting  the  position  of  the  sun  in  the 
ecliptic  when  its  apparent  diameter  is  least  or  greatest.  It  is, 
however,  very  difficult  to  tell  when  this  occurs,  as  the  apparent 
diameter  remains  very  nearly  constant  for  some  time  before 
and  after  the  earth's  passage  through  the  apse.  The  follow- 
ing is  the  method  employed: — 

When  the  earth  is  at  E  some  considerable  time  before 
the  apse  is  reached  the  sun's  apparent  diameter  is  measured 
and  its  position  in  the  ecliptic  noted  by  calculating  its  longi- 
tude. The  longitude  is  again  noted  when  its  angular  diameter 


CHAP.  VI.]  APSIDES    OF    EAKTH's    ORBIT. 


109 


measures  the  same  as  before,,  the  earth  being  at  E'.  The 
mean  of  these  two  longitudes  will  give  the  point  on  the 
ecliptic  occupied  by  the  sun  during  the  earth's  passage 
through  the  apse,  the  line  joining  this  point  to  the  earth 
giving  the  direction  of  the  line  of  apsides. 

Slow  Motion  of  the  Apse  tine.— By  observing  the 
position  of  the  apse  line  for  a  number  of  years  it  is  found 
that  it  has  a  slow  direct  motion  in  the  plane  of  the  ecliptic  at 
the  rate  of  11*25"  each  year. 

85.  Length*  of  the  Seasons. — If  P  and  Q  represent 
the  positions  of  the  earth  at  the  two  solstices,  a  perpendicular 


^ firing  ///>//, 


Autumn     / 


FIG.  45. 

XT  erected  at  8  to  PQ  (fig.  44)  will  give  the  positions 
X  and  Y  of  the  earth  at  the  vernal  and  autumnal  equinoxes 
respectively.  The  orbit  of  the  earth  is  thus  divided  into  the 
fou,  arcs  XQ,  QY,  FP,  and  PX,  corresponding  to  the  four 
seasons,  Spring,  Summer,  Autumn,  and  Winter,  respectively. 

The  four  seasons  are  unequal  in  length,  spring  and 
summer  lasting  from  21st  March  till  the  23rd  September, 
being  about  8  days  longer  than  autumn  and  winter,  which 
last  from  the  23rd  September  till  the  21st  of  the  following 
March.  This  inequality  can  very  easily  be  explained  from 
Kepler's  Second  Law,  thus : — 

Let  AB  represent  the  line  of  solstices  (fig.  45),  which  for 


110  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.  VI. 

simplicity  is  supposed  to  coincide  with  the  apse  line,  XY  being 
the  line  of  equinoxes,  and  CD  the  axis  minor  of  the  ellipse, 
i.e.  perpendicular  to  AB  erected  at  0  the  centre  of  the  ellipse 
Now  since  CD  bisects  the  area  of  the  ellipse  we  have  — 
area  CBD  =  area  CAD  ; 
area  CBD  is  >  area  XAY; 

.'.     d  fortiori  area  XB  Y  is  >  area  XA  T. 

But  since  equal  areas  are  described  in  equal  times,  it 
follows  that  the  combined  length  of  spring  and  summer  is 
greater  than  that  of  autumn  and  winter.  ri 

The  lengths  of  the  four  seasons  are  as  follows  :  — 

Spring.  Summer.  Autumn.  "Winter. 

92d20Jh.       93d14Jh.       89d18Jh.       89d  OJh. 

Eccentricity  of  the  Earth's  Orbit. 

86.  Definition.  —  The  ratio  of  the  distance  of  the  centre 
of  the  ellipse  from  the  focus  to  the  semiaxis  major  is  called 
the  eccentricity.  Thus  — 

C\  Q 

Eccentricity  E  =  —  -  (fig.  45). 
(JA 

We  can  express  the  eccentricity  of  the  earth's  orbit  in 
terms  of  the  greatest  and  least  apparent  diameters  of  the  sun. 
For,  since  08-*(8B-8A)9  ! 


_    SB-SA 
•"'          SB  +  SA 

But  SA  and  SB  are  inversely  proportional  to  the  apparent 
diameters  of  the  sun  at  A  and  B  respectively.  Therefore,  if 
d  and  d'  represent  these  diameters,  we  have 


_  d-d' 

Jo 


d  +  d' 
Therefore  the  eccentricity  of  the  earth's  orbit  is  equal  to  the 


CHAP.  VI.]  GENERAL  EXAMPLES.  Ill 

difference  between  the  greatest  and  least  apparent  diameters  of 
the  sun  divided  by  their  sum. 

Example.  —  The  greatest  apparent  diameter  of  the  sun 
being  32'  36"  and  the  least  31'  32",  calculate  from  this  the 
eccentricity  of  the  earth's  orbit. 

d-d'     1956-1892      1 


Here 


1956  +  1892      60 


GENERAL  EXAMPLES. 

1.  A  planet  is  found  to  have  an  elongation  from  the  sun  of  150°.     Is  it  an 
inferior  or  superior  planet  ?  Ans.  Superior  (Corollary,  Art.  59). 

2.  A  planet  is  found  to  be  in  quadrature.     Is  it  inferior  or  superior? 

Ans.  Superior. 

3.  Two  planets  are  observed  through  a  telescope.     One  appears  as  a  thin 
•crescent,  the  other  appears  dichotomized.     State  whether  they  are  inferior  or 
superior  planets.  Ans.  Both  inferior  (Art.  64). 

4.  If  the  exterior  angle  at  a  planet  formed  hy  lines  drawn  from  the  earth 
and  sun  he  120°,  find  what  part  of  the  hemisphere  which  is  turned  towards  the 
earth  is  illuminated.  Ans..  frd  (Art.  61). 

5.  In  question  4  find  the  ratio  of  the  apparent  hreadth  of  the  visible  illu- 
minated portion  at  its  widest  part  to  the  apparent  diameter  of  planet's  complete 
disc  (Art.  62). 

Here 

apparent  hreadth    _  r  versin  120°  _  r(l-cosl20°)  _  r  (1  +  1)  _  3r  _  3 
diameter  of  planet  2r  2r  2r       ~  4r  ~  4 

6.  Find  what   should  he  the  radius  of  a  planet's  orbit  in  order  that  its 
greatest  elongation  from  the  sun,  as  seen  from  the  earth,  should  be  30°,  assum- 
ing the  distance  of  the  earth  from  the  sun  as  92,000,000  miles. 

Ans.  46,000,000  miles. 

7.  Calculate  approximately  in  miles  per  second  the  velocity  of  the  earth  in 
its  orbit  (J.  S.,.T.  C.  D.).  Ans.  18-3. 

8.  If  there  be  378  days  between  two  successive  oppositions  of  Saturn  ;  find 
the  length  of  Saturn's  year.     (Degree,  T.  C.  D.). 

Here  -- 


and  solving,  we  get  P=  10828-6  days, 

=  29-6  vears. 


112  THE  PLANETS.       SOLAR  SYSTEM.  [CHAP.  VI. 

9.  The  periodic  time  of  Mercury  being  88  days ;  find  the  interval  between 
two  successive  inferior  conjunctions  of  this  planet. 

1  1  1 

88"  366T5  "F 
cind  solving,  we  get  T  =  115'9  days. 

10.  Assuming  the  mean  distance  of  Venus  to  he  -72,  that  of  the  earth  being^ 
unity,  apply  Kepler's  Laws  to  find  the  periodic  time  of  Venus. 

Here,  by  Kepler's  Third  Law,  we  have 

T2 :  T'2  :  :  r3  :  r'3, 
or         T2:  (365-25J2::  (-72)3  :  (1)3; 

.-.     T=  v/(365-25)3  x  (-72)3  =  223  days  nearly. 

11.  Assuming  the  distances  of  the  different  planets  from  the  sun  as  given 
by  Bode's  Law,  calculate  from  this  the  periodic  time — (1)  of  Mercury,  (2)  of 
Saturn.  Am.  (1)  90-11  days, 

(2)  11550-3  days. 

12.  Supposing  a  planet  were  to  revolve  round  the  sun  at  a  distance  of  half 
a  million  miles,  find  what  should  be  its  periodic  time. 

Am.  3|  hours  nearly. 

13.  Why  do  comets  move  with  much  greater  velocity  when  at  perihelion 
than  at  other  parts  of  their  eccentric  orbits  ?     (Kepler's  Second  Law). 

11.  The  two  satellites  of  Mars  have  periodic  times,  which  are  about  30 
hours  and  7£  hours  respectively  ;  find  the  ratio  of  their  mean  distances  from, 
Mars. 

Since  Kepler's  Laws  apply  to  the  motions  of  the  satellites,  we  have  : — 

T2  :  T*  :  :  t*  :  r'*  ; 

that  is,  (30)2  :  (7|)2  :  :  »*  :  r*, 

or  42  :  I2 :  :  r3  ;  r'3 ; 

.-.    -J/16:  1  ::  r  :  r'. 

Hence  their  mean  distances  from  Mars  are  in  the  ratio  of 
Z/IG  to  1     or    2J/2  tol. 

15.  The  velocity  of  Mercury  in  its  orbit  is  30  miles  per  second  ;  hence  calcu- 
late the  velocity  of  Saturn. 

Here,  v  :  v'  : :  V^    :  ^/r, 

or,  30  :  v'  : :  \/100:  v/4    (Bode's  Law), 

or,  30  :  v'  :  :  10  :  2  ; 

10  v'  =  60  ;     .•.  v'  =  6  miles  per  second. 

16.  Why  was  September  5,  1877,  when  the  satellites  of  Mars  were  dis- 
covered, a  date  particularly  favourable  for  observing  that  planet?    (Art.  77.) 


CHAP.  VI.]  NAMES,  PERIODS,  ETC.,  OF  THE  PLANETS.  113 


Arcs 
which  they 
retrograde. 

o             o                           o                           o             o             o             o 

•2^-2 

b          co                      ^-i                      bi         b         cb         i>- 
o             o               1            o               1            o             o             o             o 

t»              CO                                i—  *                                t-H              O<              O              I-H 

Velocity  in 
miles  per 
second. 

He«           Hot                                                                           H|n           f«N 

CO                   55                   f<                   •-!                      1 

li 

00             «O                              CO                              Tt<                              CO 

COOJOO                 |ot-(fllCO 
^H                                   1               -M              04              -*              Tj< 

1—  1 

11 

Oft                                                                                                                          |>.             10 

*O            ^                            O                            O)            CO            O)            1^- 
i—  1              OO                                00                                O4              t-              CD              <O 
^H              0                                £-                                CO              CO              CO              CO 

!i 

H*                           ^                        -HN 
OO              ^*              *O              CO              00              C^              Oi              CO              *O 
00<M«000                  li-HOIOOCO 
(N              CO              CO              CO                                                                    r-< 

Mean 
distance  from 
Sun. 

CO              (N                                C«              >O                                CO              CO              O 

cbi>-o>ocbcM»b^*o 

C/} 

xxc+e^o            5*     »c     a*     B« 

H 

S 

Illllllll 

S 

n 
r^ 


CHAPTER  VII. 

PARALLAX. 

87.  Definition. — By  the  diurnal  parallax  of  a  heavenly 
body  is  meant  the  angle  subtended  at  the  body  by  that  radius 
of  the  earth  which  is  drawn 
to  the  observer. 

Thus,  if  Cbe  the  centre 
of  the  earth,  0  the  observer, 
the  parallax  of  the  body 
M  is  the  angle  subtended 
by  CO  at  M,  viz.  the  L  p. 

The  fixed  stars  are  so 
very  far  away  that  we  may 
regard  the  lines  joining 
one  of  them  to  the  observer 
and  to  the  centre  of  the 
earth  as  being  so  nearly 
parallel  that  the  parallax  is 
practically  zero.  To  illus- 
trate how  small  this  angle  becomes,  let  the  reader  take  a 
marble  one  inch  in  diameter,  and  try  to  imagine  what 
angle  its  radius  could  subtend  at  a  point,  say  1000  miles 
away.  The  most  delicate  instrument  we  possess  would  be 
unable  to  measure  it ;  and  yet  this  angle  is  more  than  one 
hundred  times  as  great  as  the  angle  which  the  radius  of  the 
earth  could  subtend  at  even  the  nearest  fixed  star. 

The  planets,  however,  as  well  as  the  sun  and  moon,  are 
comparatively  so  near  us  that  this  difference  in  the  direction 
of  the  lines  drawn  from  a  point  on  the  surface  and  from 


Fm.  46. 


[CHAP.  VII.] 


EFFECT   OF   PARALLAX. 


115 


ithe  centre  of  the  earth  to  the  planet  is  large  enough  to  be 

(measured. 

Also  the  directions  in  which  these  bodies  are  observed 

Ifrom  any  two  positions  on  the  earth's  surface  are  not  exactly 

jthe  same.  All  observers  therefore,  wherever  situated,  reduce 
ieir  observations  to  what  they  would  be  if  situated  at  the 
mtre  of  the  earth.  This  reduction  is  what  is  called  the 

\correction  for  parallax.  The  declinations,  right  ascensions, 
;o.,  of  bodies  which  we  see  noted  in  the  Nautical  Almanac 
:e  those  which  they  would  have  if  seen  from  the  centre  of 

Ithe  earth. 

Definition. — The  horizontal  parallax  is  the  parallax  of 
body  when  on  the  horizon.     Thus,  if  the  body  M  be  on 


FIG.  47. 

the  horizon  of  the  observer  0,  the  L  P  is  the  horizontal 
[parallax. 

88.  Tbe   effect  of  Parallax  on  a  heavenly  body  is 
depress  it  in  the  heavens. 

For,  if  0  be  the  position  of  the  observer  (fig.  46),  then 
OZ,  the  production  of  the  radius  drawn  to  0,  is  the  direction 
of  the  zenith,  and  the  L  z  is  the  zenith  distar.ee  of  the  body 

i2 


116  PARALLAX.  [CHAP.  vii. 

M  as  seen  from  0.    Also  the  L  z'  would  be  its  zenith  distance 
if  the  observer  were  at  the  centre  of  the  earth  ; 

but  LZ  =  LZ'  +  LP\ 

that  is, 

apparent  zenith  distance  =  true  zenith  distance 
+  parallax  ; 

therefore,  as  seen  from  0,  the  body  appears  lower  down  in 
the  heavens  than  if  seen  from  C. 

To  find  the  parallax  of  a  body  for  a  given  zenith  distance. 

a  =  radius  of  earth  (fig.  46), 
D  -  CM=  distance  of  body. 

Since,  from  Trigonometry,  we  know  that  the  sides  of  the 
A  CO  M  are  as  the  sines  of  the  opposite  angles; 

sinp  a 

*'•    sin  (180-*)  "5 

sinp      a 
or    -T—  -=T;; 
sm  2     D 

a   . 
.v   sin  P=JJ  sins; 

but  p  being  in  all  cases  a  very  small  angle,  therefore  sin  p  =  p 
(expressed  in  circular  measure)  ; 


When  the  body  is  on  the  horizon  z  -  90°,  and  p  becomes 
the  horizontal  parallax  P  ; 


therefore  substituting,  we  have 

p  =  P  sin  z, 

or,  parallax  =  horizontal  parallax  x  sine  of  apparent  zenith 
distance. 


CHAP.  VII.]  LAW   OF   PARALLAX.  117 

Hence  tbe   parallax   of  a  heavenly  body   varies 
as  the  sine  of  its  apparent  zenith  distance. 

As  sin  s  is  a  maximum  when  a  =  90°,  we  see  that  the 
parallax  is  a  maximum  when  the  body  is  on  the  horizon. 


EXAMPLES. 

1.  Supposing  the  sun's  observed  altitude  to  be  60°  and  the  parallax  4"-4, 
find  his  true  altitude. 

Here,  since  parallax  depresses  a  body, 

true  altitude  =  observed  altitude  +  parallax  : 
therefore  true  altitude  =  60°  +  4"-4  =  60°  0'  4"-4. 

2.  Given  the  moon's    horizontal  parallax  as  being  57'  6",   find  its  true 
altitude  corresponding  to  an  observed  altitude  of  60°. 


Here  p  =  Psin«    and    s  =  900-60°  =  300; 

.-.    p  =  (57'  6")  sin  30°  =  (57'  6")£  =  28'  33"; 
therefore  true  altitude  =  60°  28'  33". 

3.  The  sun's  horizontal  parallax  being  8"-8,  find  the  true  zenith  distance 
corresponding  to  an  observed  zenith  distance  of  60°. 

Here  p  =  Psin  z, 

or     9  = 


2 
therefore         true  zenith  distance  =  60°  -  7"'6  =  59°  59'  52"-4. 

Given  the  horizontal  parallax  of  a  body,  to  find  its  distance, 
and  vice-versa. 

We  have  just  seen  that  P  =  -=;,  but  P  is  expressed  in 

circular    measure.    Hence,    if    expressed   in    seconds,    we 
have : — 


_ 
206265"  ~Z> 


118  PARALLAX.  [CHAP.  Vi 

EXAMPLES. 

1.  Given  that  the  moon's  horizontal  parallax  is  57'  6" ;  find  its  distance 
from  the  earth,  the  earth's  radius  heing  4000  miles. 

Am.  About  240,000  miles. 

2.  The  sun's  horizontal  parallax  heing  8"- 8,  find  its  distance  from  the 
earth.  Ana.  About  93,700,000  mile?. 

3.  The  moon's  distance  heing  60  times  the  earth's  radius,  find  the  moon's 
horizontal  parallax. 

P'  a       I 


Here 


206265"      D      60 ' 

.-.    P=57'17". 


89.  The  displacement  of  a  heavenly  body  due  to  parallax 
like  that  from  refraction  is  in  the  direction  of  the  vertical 
drawn  through  the  body.  Hence  the  azimuth  of  a  body  is 
not  affected  by  either  parallax  or  refraction.  "We  have  seen, 
(Art.  39)  that  refraction  does  not  depend  on  the  distance 
of  the  body  from  us,  for  the  rays  only  get  bent  on  their 
entrance  into  the  atmosphere;  the  parallax,  however,  becomes 
less  the  greater  the  distance  of  the  body,  the  moon's  hori- 
zontal parallax  being  about  57',  while  that  of  the  sun,  which 
is  much  further  away,  is  only  about  8",  and  the  fixed  stars 
are  so  remote  that  their  parallax  is  zero.  All  bodies  ex- 
cept the  moon  are  much  more  elevated  by  refraction  than 
depressed  by  parallax.  For  instance,  horizontal  refraction 
amounts  to  about  34',  whereas  the  sun  when  on  the  horizon , 
as  we  have  seen  above,  is  only  depressed  by  parallax  through 
about  8".  For  the  moon,  however,  parallax  is  much  greater 
than  refraction ;  hence  the  combined  effect  of  both  in  this 
case  produces  a  depression. 

To  find  the  Angle  which  two  distant  places  on  the  Earth's  Surface,, 
nearly  in  the  same  Meridian,  subtend  at  the  Moon  or  a 
Planet. 

90.  Let  A  and  B  be  two  distant  places  on  the  earth,  a& 
nearly  as  possible  in  the  same  meridian  (Greenwich  and  the 
Cape  of  Good  Hope  are  favourably  situated  for  the  purpose) ; 


CHAP.  VII.]    MOON  S  PARALLAX  DETERMINED.          119 

M  represents  the  moon  or  planet  when  in  the  meridian  of 
A  and  B.  Let  a  fixed  star  be  observed  from  A  and  S,  in 
nearly  the  same  part  of  the  heavens  as  M9  so  that  their  right 
ascensions  and  declinations  differ  very  slightly.  The  lines 
joining  A  and  B  to  the  star  are  nearly  parallel,  the  star 
being  so  distant. 


FIG.  48. 


The  angles  a  and  ]3,  the  angular  distances  of  the  star 
from  Jf,  are  measured  at  A  and  B  respectively  by  means  of 
micrometers;  but 


and  L  $  =  L  a  by  parallel  lines  ; 

.-.    z.0  =  Za  +  /l|3, 
and  a  and  /3  being  known,  9  is  determined. 

If  the  two  places  A  and  B  are  not  in  the  same  meridian, 
then  the  two  observers,  not  making  their  measurements  at 
the  same  time,  a  correction  must  be  made  for  the  small  dis- 
tance moved  by  the  moon  or  planet  (owing  to  the  orbital 
motion)  in  the  interval  between  its  passages  over  the  two 
meridians. 

To  find  the  Horizontal  Parallax  of  the  Moon  or  a  Planet. 

91.  Two  positions  A  and  B  (fig.  49)  are  chosen  as  before 
on  the  same  meridian  of  the  earth  in  the  northern  and 
southern  hemispheres  respectively.  The  meridian  zenith 


120  PARALLAX.  [CHAP.  vii. 

distances  of  the  moon  or  planet  M  are  then  measured 
simultaneously  by  means  of  the  meridian  circle.  Let  these 
be  z  and  2',  the  lines  O^and  OZ'  being  the  directions  of  the 


FIG.  49. 

zenith  at  A  and  B  respectively  ;  also  let  P  be  the  horizontal 
parallax.     Now  (Art.  88)  we  have — 

p  =  -=r  sin  z  =  P  sin  2, 
and  .p'  =  -=r  sins' =  P  sins'; 


.%    p  +  jf  =  P  (sin  s  +  sin  2')  ; 


sn  z  +  sn  s 


but  p  +p'  is  known,  being  angle  subtended  at  M  by  A  and  B 
(Art.  90),  and  s  and  2'  being  observed,  the  horizontal  parallax 
P  can  be  found.  This  method  is  free  from  any  serious 
errors  due  to  refraction,  for  in  Art.  90  the  moon  and  fixed 
star  are  nearly  in  the  same  position  in  the  sky,  and  therefore 
almost  equally  affected  by  refraction,  and  therefore  the  value 
of  p  +  p'  is  got  with  great  accuracy. 


CHAP.  VII.]  SUN'S   PARALLAX   DETERMINED.  121 

The  above  result  for  the  horizontal  parallax  can  be  put  in 
another  form.  For,  draw  the  equator  EQ,  and  let  /  and  t  be 
the  latitudes  of  A  and  B  respectively  ; 

.-.    L  ZOQ  =  I,  and  L  Z'OQ  =  I9 
but  (Euclid,  i.  32), 

L  Z  +  L  z  =  L  ZOM  +  L  Z  0  M  +  L  p  +  Z/, 

or  z  +  z  =  I  +  I'  +  p  +  p'  : 

.*.    p+p'  =z  +  z'  -  I-  lf\ 
z'-l-l' 


.   , 

.'.  by  substitution        P  =  - 


sin  z  +  sin  z 


It  is  not  possible  to  determine  the  sun's  parallax  in  this 
manner,  for,  owing  to  the  intensity  of  his  rays  the  neighbour- 
ing stars  cannot  be  observed.  It  can,  however,  be  calculated 
indirectly.  For,  let  the  parallax  of  Mars  when  in  opposition 
be  observed  by  the  above  method,  from  which  the  distance  of 
that  planet  from  the  earth  can  be  found  (Art.  88),  this  dis- 
tance is  the  difference  of  the  distances  (r  and  /)  of  Mars  and 
the  earth  from  the  sun  or  r  -  r'.  But  the  ratio  of  r  to  r' 
is  known  from  Kepler's  Third  Law  ;  hence  we  can  solve 
for  /  the  distance  of  the  earth  from  the  sun,  and  the  sun's 
parallax  is  determined  by  Art.  88. 

But  the  most  accurate  methods  of  obtaining  the  sun's 
parallax,  and  hence  his  distance  from  the  earth,  are  from 
observations  of  the  transit  of  Venus  across  his  disc  (Art.  76), 
as  follows  :  — 

Delisle's  Method  of  finding  the  Sun's  Parallax. 

92.  Two  stations  A  and  B  (fig.  50)  are  chosen,  both 
near  the  earth's  equator,  but  separated  as  far  apart  as  pos- 
sible, the  circle  AB  being  the  equator  of  the  earth.  Let 
us  now  suppose,  in  order  to  simplify  the  explanation,  that 
the  sun  and  the  orbit  of  Venus  VV  are  in  the  plane  of  the 
equator  AB.  Draw  tangents  AS  and  BS  to  the  sun. 


122 


PARALLAX. 


[CHAP.  vii. 


The  observer  at  A  notes  the  instant  at  which  internal 
contact  takes  place,  which  occurs  when  Venus  is  at  F, 
touching  the  line  AS  internally ;  a  similar  observation  is 
made  at  B,  internal  contact  occurring  when  Yenus  is  at  F'. 


FIG.  50. 

The  time  of  each  observation  is  reduced  to  Greenwich 
time,  which  corrects  for  the  difference  in  longitude  of  A 
and  B.  The  difference  of  the  two  results  will  give  the 
interval  of  time  during  which  Yenus  appears  to  move  round 
the  sun  through  the  angle  VSV.  (The  reader  must 
remember  that  we  are  supposing  the  earth  to  be  at  rest, 
and  that  Yenus  has  an  angular  velocity  round  the  sun  equal 
to  the  excess  of  its  real  angular  velocity  over  that  of  the 
earth.)  But  the  rate  at  which  the  angle  VS  V  is  described 
is  known,  being  360°  in  each  synodic  period :  we  can  therefore 
calculate  the  angle  VS  V  or  ASS.*  Thus,  knowing  the  angle 
which  two  distant  places  on  the  earth's  surface  subtend  at 
the  sun,  we  can  calculate  the  sun's  horizontal  parallax  as  in 
Art.  91,  and  hence  his  distance  from  the  earth. 

In  actual  practice  a  great  many  difficulties  have  to  be 
met,  the  principal  being  the  inclination  of  the  orbit  of  Yenus 
to  the  ecliptic. 


The  point  S  is  here  taken  as  practically  coincident  with  the  sun's  centre. 


CHAP.  VII.]  STINTS   PARALLAX   DETERMINED.  123 

In  Delisle's  method  the  longitudes  of  the  places  have  to 
he  very  accurately  known.  In  the  following  method,  pro- 
posed hy  Halley  in  1716,  it  is  not  necessary  to  know  the 
longitudes  of  the  places,  as  only  the  duration  of  the  transit 
is  observed  at  each  place,  the  fact  that  the  clocks  indicate 
different  hours  being  of  no  consequence. 

Halley' s  Method  or  the  Method  of  Durations. 

93.  In  this  method  the  duration  of  the  transit  is  observed 
from  two  places  A  and  B  on  the  earth,  separated  as  far 
apart  as  possible,  one  in  a  high  northern  and  the  other  in  a 
high  southern  latitude  so  that  there  may  be  as  large  a  dif- 
ference as  possible  in  the  observed  length  of  time  during 
which  the  transit  lasts,  as  seen  from  the  two  places. 


B 


FIG.  51. 

Let  V  represent  Yenus,  the  plane  containing  A,  B,  and 
V  being  the  plane  of  the  paper,  while  the  reader  must  bear 
in  mind  that  the  plane  of  the  circle,  of  which  S  is  the  centre, 
and  which  represents  the  sun's  disc,  is  at  right  angles  to  this 
plane.  To  an  observer  at  A,  Venus,  in  her  apparent  motion 
in  the  direction  indicated  by  the  arrow,  will  appear  to  cross 
the  sun's  face  in  the  direction  cd9  and  to  the  observer  at  B  in 
the  direction  ab,  the  time  occupied  being  noted  in  each  case. 
But  the  rate  at  which  Venus  appears  to  cross  the  sun's  face 
can  be  calculated  (Ex.  4,  p.  125),  being  at  the  rate  of  4"  in  each 
minute  of  time,  we  can  therefore,  by  a  statement  in  simple 
proportion,  calculate  the  number  of  seconds  in  ab  and  cd 


124  PARALLAX.  [CHAP.  VII. 

respectively,  very  much  more  accurately  than  if  measured  by 
a  micrometer.  Therefore  the  halves  of  these  chords,  viz.  eb 
and  fd  are  known  in  seconds.  But  the  sun's  semidiameter 
sb  or  sd  is  also  known  in  seconds  ;  therefore  we  can  find  the 
number  of  seconds  in  se  and  sf,  for  we  have  approximately, 

by  Euclid  (I.  47), 

se*  =  sb*-bez; 

also  sfz=s(^-dfz. 

Knowing  se  and  s/,  we  find,  by  subtraction,  the  number 
of  seconds  in  ef. 

Again,  the  number  of  miles  in  efcan  be  found  from  know- 
ing the  number  of  miles  between  the  two  stations  A  and  B, 
for,  regarding  the  two  triangles  A  VB  and  eVf  as  similar,  we 
have 


But  the  ratio  of  F/to  AV  can  be  found  (Art.  66), 
being  723  to  277  ;  therefore  ef  in  miles  is  known.  Lastly, 
knowing  efin  miles  and  the  angle  in  seconds  subtended  by  it 
at  the  earth,  the  distance  of  the  sun  can  be  found  thus  :  — 

ef"  efin  miles 

206265  =  sun's  distance' 

and  consequently  his  parallax  is  determined. 

To  find  the  Radius  of  the  Moon  in  Miles. 
94.  Having  shown  how  to  determine  the  parallax  of  the 


FIG.  62. 
moon,  sun,  or  a  planet,  we  can  calculate  the  radii  of  these 


CHAP.  VII.]  MAGNITUDE  OF  MOON  OR  PLANET  DETERMINED.  125 

bodies  in  miles  by  a  comparison  with  the  radius  of  the 
earth. 

» 

Let  r  =  radius  of  earth. 

/  =  radius  of  moon,  or  other  body. 

P  =  moon's    horizontal  parallax  =  earth's  angular 
semidiameter  as  seen  from  the  moon. 

P*  =  moon's  angular  semidiameter. 
Now  -3  =  P  (in  circular  measure), 

-  =  P  (in  circular  measure) ; 


or   (radius  of  earth)  :  (radius  of  moon)  :  :  (moon's 
parallax)  :  (moon's  semidiameter). 

EXAMPLES. 

1.  Taking  the  moon's  horizontal  parallax  as  57',  and  its  angular  diameter 
as  32',  find  its  radius  in  miles,  assuming  the  earth's  radius  to  he  4000  miles. 

Here  moon's  semidiameter  =  16' ; 

.-.    4000  :  /  :  :  57'  :  16' ;  .-.  r'  =  400°  *  16  =  1123  miles. 

&7 

2.  The  sun's  horizontal  parallax  heing  8"-8,  and  his  angular  diameter  32', 
find  his  diameter  in  miles.  Ans.  872,727  miles. 

3.  The  synodic  period  of  Venus  heing  584  days,  find  the  angle  gained  in 
each  minute  of  time  on  the  earth  round  the  sun  as  centre. 

Ans.  l"-54  per  minute. 

4.  Find  the  angular  velocity  with   which  Venus  crosses  the  sun's  disc, 
assuming  the  distances  of  Venus  and  the  earth  from  the  sun  are  as  7  to  10,  as 
given  by  Bode's  Law. 

Since  (fig.  50)  8V:  VA  :  :  7 :  3.  But  /SThas  ^.relative  angular  velocity  round 
the  sun  of  1"'54  per  minute  (see  Example  3) ;  therefore,  the  relative  angular 
velocity  of  AT  round  A  is  greater  than  this  in  the  ratio  of  7  :  3,  which  gives  an 
approximate  result  of  3"-6  per  minute,  the  true  rate  heing  about  4"  per  minute. 


126  PARALLAX.  [CHAP, 


Annual  Parallax. 

95.  We  have  already  seen  that  no  displacement  of  the 
observer  due  to  a  change  of  position  on  the  earth's  surface 
could  apparently  affect  the  direction  of  a  fixed  star.  How- 
ever, as  the  earth  in  its  annual  motion  describes  an  orbit  of 
about  92  million  miles  radius  round  the  sun,  the  different 
positions  in  space  from  which  an  observer  views  the  fixed 
stars  from  time  to  time  throughout  the  year  must  be  sepa- 
rated from  one  another  by  very  great  distances  indeed. 
For  instance,  any  two  diametrically  opposite  points  in  the 
orbit  of  the  earth  are  separated  by  an  interval  of  about  184 
million  miles,  the  earth  proceeding  from  one  of  these  points 
to  the  other  in  about  six  months.  "We  should  therefore 
expect  that,  when  viewed  from  two  points  separated  by 
such  a  distance  as  this,  the  fixed  stars  should  not  occupy 
exactly  the  same  position  with  respect  to  one  another, 
those  which  are  nearer  the  earth  being  more  displaced  than 
those  further  away.  This  is  to  a  certain  extent  quite  true  ; 
but  to  such  vast  depths  are  the  fixed  stars  sunk  in  space, 
that  only  in  the  case  of  some  of  those  nearest  to  us  can  any 
appreciable  displacement  be  detected ;  or,  in  other  words,  a 
base  line  of  184  million  miles  is  much  too  small  a  distance  to 
take  in  an  attempt  to  measure  the  distances  of  by  far  the 
greater  number  of  these  bodies. 

Owing  to  the  small  displacements  in  the  apparent  direc- 
tions of  some  of  the  fixed  stars,  due  to  the  earth's  changes 
of  position  throughout  the  year,  we  refer  their  directions  on 
the  celestial  sphere  to  what  they  would  have  if  viewed  from 
the  centre  of  the  sun,  which  is  fixed.  This  direction,  as 
seen  from  the  centre  of  the  sun,  being  called  the  star's  helio- 
centric direction,  the  correction,  which  must  be  applied  to 
reduce  the  apparent  or  geocentric  to  the  heliocentric  direction 
being  called  the  correction  for  annual  parallax. 


/UNIVERSITY) 

V  or  J 

VJ^LIFORNV'W' 


CHAP.  VII.]  ANNUAL   PARALLAX.  127 

Definition. — The  annual  parallax  of  a  star  is  the  angle 
subtended  at  the  star  by  the  line  joining  the  earth  and  sun. 
Thus,  if  E  represent  the  earth,  E  the  sun,  and  8  a  star,  the 
annual  parallax  of  8  is  the  angle  subtended  at  8  by  EH  or 
the  angle/?. 


96.  The  law  according  to  which  the  annual  parallax  of  a 
star  should  vary  can  be  deduced  by  a  method  similar  to  that 
applied  to  the  diurnal  or  geocentric  parallax  of  the  moon  or 
planets.  For 

sin  p     r 

smE=d' 

T 

.*.     sin^=-sin^; 
but^>  being  small,  sin^>  =p  (in  circular  measure), 

j0=-sin-E': 
a 

therefore,  the   annual  parallax  varies  as  the  sine  of  the 
angular  distance  of  the  sun  from  the  star. 

It  is  evident  that  the  parallax  of  a  star  is  a  maximum 
when  E  =  90°,  which  happens  twice  a  year  for  each  star. 
Let  P  represent  this  maximum  value,  and  we  have — 

P  =  -sin  90°=^;  alsop  =  P  sin  E. 

Ct  €v 

N.B. — G-enerally  speaking,  by  a  star's  parallax  is  meant 
this  maximum  value  of  the  parallax,  unless  otherwise  stated. 


128  PARALLAX.  [CHAP.  VII. 

As  P  is  expressed  in  circular   measure  in  the  above 
formula;  therefore,  when  expressed  in  seconds,  we  have — 

P"         r 

206265  "  d 

Thus,  when  the  parallax  of  a  star  is  known,  we  can  deduce 
its  distance  from  the  solar  system  as  r  the  distance  of  the 
earth  from  the  sun  is  known  (Art.  92).  In  the  following 
questions,  r  may  he  taken  as  92  million  miles. 

EXAMPLES. 

The  parallax  of  a  Centauri  being  0"-8,  find  its  distance  from  the  solar 
system. 

-8"     _  92000000 
206265  "        d        ' 

92000000  x  206265 
.•.    d  = - miles. 

*0 

2.  Supposing  the  parallax  of  a  star  to  be  0"'2,  find  how  long  a  ray  of  light 
would  take  to  travel  to  the  earth,  being  given  the  velocity  of  light  as  190,000 
miles  per  second.  Ans. 


The  effect  of  annual  parallax  on  a  star  is  to 
cause  it  to  appear  to  move  in  a  small  ellipse 
throughout  the  year. 

97.  For  as  each  displacement  of  the  earth  in  its  orhit 
produces  a  corresponding  small  displacement  in  the  appa- 
rent position  of  the  star  in  the  sky,  the  star  will  therefore 
seem  to  describe  a  small  yearly  orbit  round  its  heliocentric 
position  (which  is  fixed)  parallel  to  the  plane  of  the  earth's 
orbit.  If  now  we  assume  the  earth's  orbit  to  be  circular,  we 
will  consider  the  effect  of  parallax  on  a  star,  according  as  it 
is  situated — (1)  near  the  pole  of  the  ecliptic,  (2)  on  the 
ecliptic,  (3)  at  any  point  of  the  sky. 

(1)  If  a  star  be  situated  in  the  pole  of  the  ecliptic,  the  plane 
of  the  small  arc  which  we  may  suppose  it  to  describe  being 
at  right  angles  to  our  line  of  sight  will,  when  projected  on 
the  celestial  sphere,  still  appear  circular. 


CHAP.  VII.]       ANNUAL    PARALLAX.       BESSEL*&   METHOD.  129 

(2)  If  situated  on  the  ecliptic,  it  will  appear  to  move  back 
and  forward  along  the  ecliptic  in  a  straight  line,  this  being 
explained  by  the  fact  that  a  circle  seen  edgeways  appears 
as  a  line. 

(3)  If  situated  in  any  other  part  of  the  sky,  the  apparent 
path  throughout  the  year  will  appear  as  a  diminutive  ellipse, 
as  a  circle  seen  obliquely  will  appear  elliptic. 

To  determine  the  Annual  Parallax  of  a  Star — Bessel's 
Method. 

98.  Bessel's  method,  otherwise  called  the  differential 
method,  consists  in  choosing  a  very  faint  star  very  close  to 
the  star  whose  parallax  is  sought.  Being  very  faint,  it  is 


FIG.  54. 


presumably  much  further  away  than  the  star  in  question ; 
and  we  may,  therefore,  assume  that  its  own  parallax  is  so 
very  small  that  any  changes  which  take  place  throughout 
the  year  in  the  angular  distance  of  the  two  stars  from  one 
another  must  be  due  almost  entirely  to  the  parallax  of  the 
near  one.  The  actual  measurement  of  these  changes  enables 
us  to  determine  the  annual  parallax. 

N.B.—  The  faint  star  is  chosen   very  close  to  the  star, 


130  PARALLAX.  LCHAP-  V11' 

whose  parallax  we  want,  in  order  that  both  may  be  equally 
affected  by  errors  due  to  refraction,  aberration,  &c.,  so  that 
we  have  not  to  correct  for  these  errors.  The  following  will 
illustrate  the  method  employed  :  — 

Let  A  and  B  be  two  diametrically  opposite  points  in  the 
earth's  orbit.  AS  and  BS(ftg.  54)  are  the  directions  of  the 
star  S,  as  viewed  from  A  and  B\  and  AS'  and  BSf  the  direc- 
tions of  the  faint  star,  these  lines  being  taken  as  parallel, 
owing  to  the  much  greater  distance  of  S'.  A,  B,  S,  and  S' 
are  supposed  in  the  same  plane. 

At  A  the  observer  measures,  by  means  of  the  micrometer 
or  heliometer,  the  angle  a  between  S  and  $';  and  again  at  B, 
six  months  afterwards,  he  measures  the  angle  |3.  But  by 
Euclid  (i.  32),  we  have  — 


but  L  0  =  L  |3  by  parallel  lines  ; 


but  a  and  |3  are  known,  therefore  6  is  determined  ;  and 
6  being  the  angle  subtended  by  the  diameter  of  the  earth's 
orbit,  is  twice  the  annual  parallax  which  can  therefore  be 
found. 

N.B.  —  The  reader  can  compare  this  method  with  that 
employed  to  find  the  angle  subtended  at  the  moon  by  two 
distant  places  on  the  earth's  surface  (Art.  90)  by  means  of 
which  the  moon's  diurnal  or  geocentric  parallax  was  deter- 
mined. 

Strictly  speaking,  the  lines  AS'  and  BS'  have  some 
inclination  to  each  other;  therefore,  the  error  to  which 
Bessel's  method  is  open  is,  that  it  determines  not  the 
parallax  of  the  near  star,  but  the  difference  in  parallax 
of  the  two  stars  ;  hence  the  parallax  of  a  star  thus  deter- 
mined is  always  too  small,  but  never  too  great,  and 


CHAP.   VII.]         ANNUAL    PARALLAX    OF   JUPITER.  131 

therefore  the  distance  of  a  star  from  us  is  found  to  be 
greater  than  is  actually  the  case. 

Bessel,  by  this  method,  first  measured  the  parallax,  and 
hence  the  distance  of  61  Cygni  in  the  year  1838,  and  in 
the  following  year  that  of  a  Centauri  was  found. 

Instead  of  measurements  with  the  micrometer,  photo- 
graphy is  now  being  very  successfully  employed  in  noting 
the  changes  in  the  angular  distance  of  the  two  selected 
stars  from  one  another. 

Absolute  Method. 

99.  This  method  consists  in  measuring  the  star's  right 
ascension  and  decimation  when  in  the  meridian  at  different 
times  throughout  the  year,  and  after  making,  with  as  much 
accuracy   as    possible,    all   the    corrections    for   precession, 
nutation,  &c.,  the  different  results  are  compared  together 
when  any  small  differences   which  they  may   show   give 
sufficient  data  to  calculate  the  annual  parallax  of  the  star. 

EXAMPLES. 

1.  («)  Where  must  a  star  be  situated  so  as  to  have  no  displacement  due 
to  parallax  ;  (b)  where  must  it  be  situated  so  that  the  effect  of  parallax  may  be 
greatest. 

Ans.     (a)   In  a  line  with  earth  and  sun. 

(b)    At  an  angular  distance  of  90°  from  sun. 

2.  If  the  parallax  of  61  Cygni  be  0"'5,  find  the  parallax  of  a  star  which  is 
ten  times  as  far  away  from  our  solar  system.  Ans.  0"'05. 

3.  The  parallax  of  a  Centauri  being  0"'75,  compare  its  distance  with  that 
of  61  Cygni,  whose  parallax  is  0"'5. 

Ans.    (dist.  a  Centauri)  :  (dist.  61  Cygni)  :  :  2  :  3. 

To  find  the  Annual  Parallax  of  Jupiter. 

100.  As  the  distance  of  Jupiter  at  opposition  is  more  than 
four  times  as  great  as  that  of  the  sun,  its,  diurnal  parallax 
is  therefore  very  small,  so  that  it  is  not  possible  to  observe 

K2 


132  PARALLAX.  [CHAP.  VII. 

it  with  the  same  degree  of  accuracy  as  in  the  case  of  Mars 
(Art.  91).  Its  annual  parallax,  however,  may  be  found 
thus : — 

Let  $,  E,  and  J  represent  the  sun,  earth,  and  Jupiter, 
respectively  when  Jupiter  is  in  quadrature,  i.e.  when  the 
angle  SEJ  is  a  right  angle.  Again,  let  $,  Ef,  J'  be  their 
positions  when  Jupiter  is  again  in  quadrature,  the  earth 
having  moved  through  Eff  and  Jupiter  through  J>F. 


FIG.  55. 

The  number  of  days  between  the  two  observations  being 
known,  we  therefore  know  the  LESE'  described  by  the 
earth  in  that  time.  Similarly,  we  know  the  L  JSJ'.  There- 
fore, the  /.a,  which  is  half  the  difference  of  these  two  angles 
(the  two  triangles  SEJ  and  SE'J'  being  equal  in  every 
respect),  is  known. 

But  the  angle  a  is  the  complement  of  the  L  9.  Therefore 
0  is  found  ;  that  is,  the  angle  subtended  at  Jupiter  by  the 
radius  of  the  earth's  orbit  is  known. 

Again,     «n 


which  determines  the  distance  of  Jupiter. 

This  method  also  applies  also  to  any  planet  outside  the 
orbit  of  Jupiter. 

It  can  be  easily  shown  that  if  T  represent  the  synodic 


CHAP.     VII.]  EXAMPLE.  133 

period  of  Jupiter,  and  Q  the  interval  between  its  eastern  and 
western  quadratures,  the  annual  parallax 


90C 

for     ^——^  =  £  gained  by  earth  on  Jupiter  in  Q  days ; 

180°Q 


,fl      ,~9 
.'.  (fig.  55)  2a  = 


«  = 


180°£)  /       2O\ 

.-.  annual  parallax  6  =  90°  -  -^-^  =  90°f  1  -^  V 


EXAMPLE. 

The  interval  between  eastern  a-.id  western  quadratures  of  Jupiter  is  175  days, 
and  between  two  oppositions  400  days  ;  find  the  annual  parallax  of  this  planet. 

Am.  11°  15'. 


(    134    ) 


CHAPTER  VIII. 

DETERMINATION  OF  THE  FIRST  POINT  OF  ARIES.        PRECESSION, 
NUTATION,   AND  ABERRATION. 

101.  The  first  point  of  Aries  being  the  zero  point  from 
which  the  right  ascensions  of  all  heavenly  bodies  are  mea- 
sured, it  is  therefore  necessary  to  know  its  position  with 
reference  to  the  fixed  stars  with  very  great  accuracy.  Once 
having  fixed  this  point,  and  the  astronomical  clock  being 
set  at  zero  when  it  crosses  the  meridian,  then  the  time  at 
which  any  other  star  crosses  the  meridian  will,  on  being 
reduced  to  degrees  (by  allowing  15°  for  each  hour),  give 
the  right  ascension  of  that  star. 

It  is  evident  that  if  we  could  find  independently  the 
right  ascension  of  any  one  star,  the  position  of  the  first 
point  of  Aries  is  immediately  determined,  and,  conse- 
quently, the  right  ascensions  of  all  other  stars.  The 
following  method  was  first  used  by  Flamsteed,  the  star 
selected  being  a  Aquilse. 

Flamsteed9  s  method  of  finding  the  Right  Ascension  of  a  Star. 

Let  a  be  the  star  whose  right  ascension  is  sought  (fig.  56) : 
we  have,  therefore,  to  find  X  T  (X  being  the  foot  of  the 
declination  circle  drawn  through  a). 

The  declination  of  the  sun  SM  is  measured  (Art.  34)  at 
noon  on  some  day  shortly  after  the  vernal  equinox,  this 
being  done  by  measuring  his  meridian  zenith  distance.  At 
the  same  time  the  interval  between  his  transit  across  the 
meridian  and  that  of  the  star  a  is  noted,  this  interval  being 
the  difference  of  their  right  ascensions  MX,  which  we  will 
denote  by  a. 


CHAP,  viii.]  FLAMSTEED'S  METHOD.  135 

Again,  it  can  be  ascertained  at  what  time  the  sun  shall 
have  an  equal  declination  shortly  before  the  Autumnal 
Equinox  by  observing  his  meridian  zenith  distance  at  noon 
on  successive  days  previous  to  September  23rd.  But  here 
we  will  have  to  do  a  little  calculation  ;  for  it  is  very  impro- 
bable that  the  sun  will  have  an  equal  declination  exactly 
at  noon  on  any  one  of  these  days ;  but  we  can  observe  his 
declination  at  noon  on  two  successive  days,  at  which,  in  one 
case,  it  is  greater,  and,  in  the  other  case,  less  than  SM ;  and 


-Equator 


FIG.  56. 


then,  if  we  assume  that  for  short  periods  his  changes  in 
right  ascension  and  declination  are  proportional  to  one 
another,  we  can,  by  a  simple  statement  in  proportion,  calcu- 
late the  exact  time  at  which  his  declination  SiN"  shall  be 
equal  to  SM.  In  this  case  also  the  difference  between 
his  right  ascension  and  that  of  the  star  a  is  noted.  This 
difference  is  NX,  which  we  will  call  |3.  It  is  evident  that 


Let        x  -  the  required  right  ascension  T  X  of  star, 
fj.  =  right  ascension  of  sun  at  S  =  MV  ; 
/.     180°  -  fj.  =  right  ascension  of  sun  at  Si  = 
but  XT  -  jffr  -  MX, 

or  x  -  n      =  a  ; 

also  x-  (lSO°-fjL)  =/3,     or    ^-180°  +  w  =  6. 


136  PRECESSION,  NUTATION,   ABERRATION.          [CHAP.  V11I- 

Thus   we  have   two  simultaneous  equations,  the  unknown 
quantities  being  x  and  /x.     Adding,  we  get 


but  a  and  |3  being  known,  x  is  therefore  determined. 

The  above  formula  is  open  to  some  error,  as  during  the 
interval  between  the  two  observations  there  is  a  slight 
increase  in  the  right  ascension  of  the  star,  owing  to  pre- 
cession. It  can,  however,  be  corrected  as  follows  :  — 

Let  p  =  the  increase  in  the  R.A.  of  a  star  dining  the 
interval,  and  our  two  equations  become 

x  -  p  =  a, 
and  x+p- 


The  uncorrected  value  of  x  gives,  not  the  star's  right 
ascension  during  the  first  observation  near  the  vernal 
equinox,  but  its  mean  value  between  the  two  observations 
or  the  value  it  would  have  at  about  the  21st  June. 

The  advantages  of  Flamsteed's  method  are  that  it 
is  not  necessary  to  know  exactly  the  sun's  declination  ; 
it  is  quite  sufficient  to  observe  when  the  declinations 
at  the  two  observations  are  equal,  so  that  any  uncer- 
tainty in  the  latitude  of  the  place  due  to  instrumental 
errors,  which  will  affect  both  observations  equally,  is  of 
no  consequence.  Also,  as  the  sun  has  nearly  the  same 
zenith  distance  during  each  observation,  he  will  be  equally 
affected  by  refraction  and  parallax,  and  hence  these  errors 
are  avoided. 


CHAP.  VIII.]      OBLIQUITY  OF  EL1PTIC  TO  EQUATOR.  137 

Determination  of  the  Obliquity  of  the  Ecliptic  to  the 
Equator. 

102.  This  angle  is  measured  by  observing  the  meridian 
zenith  distances  of  the  sun  at  the  summer  and  winter  solstices. 
Let  these  be  z  and  2',  and  let  the  latitude  of  the  place  be  /. 
Now  if  S  be  the  position  of  the  sun  at  one  of  the  solstices, 
its  declination  SM  is  equal  to  the  inclination  w  of  the 
ecliptic  to  the  equator  (fig.  57),  for  the  angle  between  two 
great  circles  is  measured  by  the  arc  they  intercept  on  a 
circle  perpendicular  to  both. 


FIG.  57. 
But  latitude  =  zenith  distance  ±  declination*  (Art.  34)  ; 

.*.     I  =  z  +  o>  for  summer  solstice, 
also  ?  =  z'  -  M  for  winter  solstice. 


Subtracting,  we  get          u>  = 

/ 

and,  therefore,  cu  is  found. 

In  the  above  observations,  it  is  not  probable  that  the 
sun  will  be  exactly  at  the  solstitial  point  when  in  the 
meridian,  but  allowance  can  be  made  for  his  change  of 
declination  during  the  interval. 

*  This  equation  is  obviously  identical  with  that  given  in  Art.   34,  viz. 
colat  ±  8  =  o. 


138  PRECESSION,  NUTATION,  ABERRATION.       [CHAP.  VIII. 

Precession  of  the  Equinoxes. 

103.  Eepeated  observations  of  the  right  ascensions  and 
decimations  of  the  stars  extending  over  a  long  period  of 
time  show  us  that  the  first  point  of  Aries  is  not  a  fixed 
point  in  the  sky,  but  has  a  very  slow  movement  among 
the  fixed  stars  along  the  ecliptic  in  a  direction  opposite  to 
that  of  the  yearly  motion  of  the  sun.  This  backward 
motion  of  Aries  to  meet  the  sun,  in  which  the  first  point 
of  Libra  also  takes  part,  causes  the  equinoxes,  as  it  were, 
to  precede  their  due  time  each  year.  Hence  this  slow 
movement  is  called  the  precession  of  the  equinoxes. 

The  rate  of  precession  is  50"*24  in  one  year,  or  about 
1°  in  72  years.  The  time  taken  by  Aries  to  complete  one 
revolution  of  the  heavens  would,  therefore,  be  about  26,000 
years  ;  for 

360°  x  60  x  60      _ 

=      '        JearS  y' 


Owing  to  precession,  the  longitude  of  each  fixed  star 
increases  at  the  rate  of  50"*24  each  year.  The  right  ascen- 
sions and  declinations  of  the  stars  are  also  found  to  be 
slowly  changing,  but  their  latitudes  remain  almost  constant. 
From  this  latter  consideration  we  are  led  to  the  con- 
clusion that  the  ecliptic  is  very  nearly  fixed  in  the  heavens, 
but  that  the  equator  must  be  slowly  shifting  on  the  ecliptic, 
thus  causing  their  intersections  T  and  ===  to  move  in  the 
manner  described  above. 

This  movement  of  the  equator  on  the  ecliptic  is  accom- 
panied by  a  corresponding  gradual  displacement  of  the 
celestial  pole,  which  describes  a  circular  path  round  the 
pole  of  the  ecliptic  at  a  distance  from  it  of  23°  28',  one 
revolution  being  completed  in  26,000  years.  Hence,  in 
some  thousands  of  years  the  star  which  is  now  our  pole 
star  will  be  at  a  considerable  distance  from  the  celestial 


(HAP.  VIII.]  CAUSES    OF    P  RECESSION.  139 

pole.  The  bright  star  a  Lyrse  will,  in  about  10,000  years, 
be  distant  about  5°  from  the  point  round  which  the  heavens 
will  then  seem  to  revolve,  and  will,  like  our  present  pole 
star,  appear  almost  stationary  in  the  sky. 


Physical  Cause  of  Precession. 

104.  The  precession  of  the  equinoxes  is  almost  entirely 
caused  by  the  attraction  of  the  moon  and  sun  on  the  pro- 
tuberant portions  of  the  earth  at  the  equator^  If  the  earth's 
shape  were  perfectly  spherical,  the  attractions  of  the  sun 
and  moon  could  each  be  represented  by  a  single  force 
passing  through  its  centre,  and  would,  therefore,  not  disturb 
the  axis  of  rotation  of  the  earth  nor  the  plane  of  the 
equator.  However,  the  shape  of  the  earth  is  spheroidal, 
not  unlike  that  of  a  sphere  with  an  additional  layer  or  belt 
of  matter  round  the  equatorial  regions.  In  the  adjoining 
figure,  let  8  represent  the  sun,  and  PPf  the  axis  of  rotation 
of  the  earth  (fig.  58). 


B 


5^0.  58 

Now,  the  attraction  of  the  sun  on  the  protuberant  por- 
tions of  the  earth  being  greater  on  the  nearer  than  on  the 
more  remote  side,  the  resultant  attraction  will,  therefore, 
be  represented  by  a  single  force,  OB  acting  at  a  point  0 
above  the  centre  of  gravity  C,  the  effect  of  which  would 
be  to  cause  a  disturbance  of  the  axis  of  rotation  of  the 
earth.  Qn  first  thoughts  we  might  imagine  that  this  would 


140  PRECESSION,    NUTATION,    ABERRATION.       [CHAP.  VIII. 

result  in  a  change  in  the  plane  of  the  equator,  so  as  to 
make  it  eventually  coincide  with  the  ecliptic,  and  set  the 
earth's  axis  at  right  angles  to  the  plane  of  the  ecliptic. 
And  certainly  this  would  be  the  case  were  it  not  that  the 
earth  is  at  the  same  time  rotating  rapidly  round  its  axis, 
the  resultant  effect  of  these  two  rotations  being  that  the 
axis  of  the  earth  is  indeed  disturbed,  but  in  such  a  manner 
as  not  to  alter  its  angle  of  inclination  to  the  plane  of  the 
ecliptic.  In  fact,  the  axis  of  the  earth  has,  as  it  were, 
a  slow  "  wobbling  "  motion,  so  that  the  point  in  the  heavens 
to  which  it  is  directed,  viz.  the  celestial  pole,  describes  the 
circle  round  the  pole  of  the  ecliptic,  which  we  have  pre- 
viously mentioned. 


FIG.  59. 

This  motion  of  the  earth's  axis  can  be  very  well  illus- 
trated by  the  "wobbling"  of  the  axis  of  rotation  of  a 
spinning-top.  The  weight  of  the  top  acting  vertically 
downwards  tends  to  pull  the  axis  of  rotation  AB  away  from 
the  vertical ;  but  if  the  top  be  spinning  sufficiently  rapidly, 
it  will  not  fall  to  the  ground,  but,  as  we  all  know,  the  axis 
of  rotation  describes  a  cone  round  the  vertical  AO,  keeping 
at  a  constant  angle  to  the  ground  in  precisely  the  same 


CHAP.  VIII. J  NUTATION.  141 

manner  as  in  the  case  of  the  earth  the  celestial  pole,  which 
is  the  extremity  of  its  axis,  revolves  round  the  pole  of  the 
ecliptic. 

The  disturbing  effect  of  the  moon's  attraction  is  more 
than  twice  as  great  as  in  the  case  of  the  sun,  the  ratio 
being  as  7 :  3,  the  reason  of  this  being  on  account  of  the 
greater  proximity  of  the  moon  to  the  earth. 

In  either  case,  the  disturbance  is  greatest  when  the 
attracting  body  reaches  its  greatest  north  or  south  decli- 
nation, and  is  zero  when  the  body  is  on  the  celestial  equator. 

The  precession  caused  by  the  sun  and  moon  is  some- 
times called  the  lunar-solar  precession.  It  amounts  to  50"*35 
annually.  This  lias,  however,  to  be  diminished  by  a  very 
small  amount  called  the  planetary  precession,  which,  acting 
in  the  opposite  direction,  is  found  to  be  0"-11  each  year, 
and  leaves  an  annual  general  precession  of  50"'24.  The 
planetary  precession  is  caused  by  the  action  of  the  planets, 
which  tends  to  disturb  the  earth's  orbit,  and  therefore 
the  plane  of  the  ecliptic,  producing  at  the  same  time  a 
diminution  of  the  obliquity  of  the  ecliptic  to  the  equator 
of  about  half  a  second  each  year.  However,  this  change 
of  obliquity  will  never  exceed  a  certain  fixed  limit,  viz. 
about  1  y  on  either  side  of  the  mean  value. 

At  present  the  vernal  equinoctial  point,  though  still 
retaining  the  name  "  First  point  of  Aries,"  is  not  in  the 
constellation  of  Aries,  but,  owing  to  precession,  has  shifted 
about  30°  into  the  neighbouring  constellation  Pisces.  Also 
the  autumnal  equinoctial  point  is  not  now  in  the  constellation 
of  Libra  but  in  Yirgo. 

Nutation. 

105.  So  far  we  have  dealt  with  precession  as  if  the 
celestial  pole  moved  uniformly  in  a  circle  round  the  pole  of 
the  ecliptic,  and  this  would  certainly  be  the  case  if  the 
disturbance  due  to  the  attractions  of  the  sun  and  moon  were 


142  PRECESSION,    NUTATION,    ABERRATION.      [CHAP.  VIII 

constant ;  but  in  consequence  of  a  want  of  uniformity  in  this 
disturbance  the  celestial  pole  really  describes  a  wavy  path  (see 
fig.  60).  This  nodding,  as  it  were,  of  the  celestial  pole  to 
and  from  the  pole  of  the  ecliptic  is  called  nutation.  The 
result  is,  that  the  precession  is  sometimes  more  and  at  other 
times  less  than  its  mean  value,  and  there  also  results  a  small 
periodic  increase  and  diminution  of  the  obliquity  of  the 
ecliptic  to  the  equator,  according  as  the  celestial  pole  P 
approaches  or  recedes  from  the  pole  of  the  ecliptic. 


FIG.  60. 

Nutation  is  almost  altogether  caused  by  the  variable 
action  of  the  moon  depending  on  the  position  of  the  moon's 
nodes  (the  points  where  its  path  cuts  the  ecliptic)  which  make 
a  complete  revolution  of  the  heavens  in  18f  years. 

The  wavy  motion  of  the  celestial  pole  may  be  graphically 
represented  in  the  following  manner  : — 

Bound  the  mean  position  of  the  celestial  pole  as  centre 
(fig.  60)  describe  a  small  ellipse  ab,  with  a  major  axis 
ab  =  18"-5  directed  towards  the  pole  of  the  ecliptic,  and  a 
minor  axis  13"*7  along  the  circle  mn  ;  then  if  we  imagine  the 


CHAP.  VIII.]          VELOCITY  OF  LIGHT.       ABERRATION.  143 

mean  pole  which  is  the  centre  of  the  ellipse,  to  move  along 
the  arc  mn,  the  true  pole  P  will  move  in  the  ellipse  round 
it  as  centre,  completing  a  revolution  in  18f  years. 

Bradley  first  discovered  nutation  by  observing  that,  after 
correcting  for  aberration,  &c.,  the  apparent  displacements 
in  the  fixed  stars  with  reference  to  the  equator  and  ecliptic 
could  not  be  accounted  for  on  the  supposition  of  a  uniform 
precession. 

The  Velocity  of  Light. 

106.  That  the  propagation  of  light  is  not  instantaneous  was 
discovered  by  Eoemer,  in  1675  from  observing  the  eclipses  of 
Jupiter's  satellites.  The  times  at  which  these  eclipses  should 
occur  were  predicted  from  a  great  number  of  previous  obser- 
vations, and  would  therefore  correspond  to  the  mean  distance 
of  the  planet  from  the  earth.  It  was  found,  however,  that 
when  Jupiter  was  in  opposition,  or,  in  other  words,  nearest  to 
the  earth,  the  eclipses  appeared  to  occur  about  eight  minutes 
before  the  calculated  time.  On  the  other  hand,  when  Jupiter 
was  in  superior  conjunction  or  furthest  from  the  earth  the 
observed  time  was  about  eight  minutes  later  than  that  pre- 
dicted ;  from  which  it  appeared  that  this  difference  of  about 
sixteen  minutes,  or,  more  accurately,  sixteen  minutes  and 
thirty- six  seconds,  was  the  time  taken  by  a  ray  of  light  to 
move  through  the  diameter  of  the  earth's  orbit.  Taking 
this  distance  as  185,000,000  miles  we  get  that  the  velocity  of 
light  is  about  186,000  miles  per  second.  As  the  velocity  of 
the  earth  in  its  orbit  is  18 J  miles  per  second,  we  see  that  the 
velocity  of  light  is  about  10,000  times  greater  than  that  of 
the  earth. 

We  see  from  the  above  that  light  takes  about  8m  18s  to 
pass  from  the  sun  to  the  earth.  This  interval  is  sometimes 
called  the  equation  of  light.  The  velocity  of  light  has  since 
been  measured  directly  by  M.  Fizeau,  and  afterwards  by 
M.  Foucault. 


144  PRECESSION,  NUTATION,  ABERRATION.  [CHAP.  VIII. 

Aberration. 

107.  The  Aberration  of  Light  is  the  apparent  displace- 
ment in   the   directions  of  the  heavenly  bodies  due  to  a 
combination  of  the  velocity  of  the  earth  with  that  of  light. 
The  velocity  of  the  earth,  although  small  compared  with  that 
of  light,  is  still  large  enough  to  produce  a  sensible  deflection 
in  the  direction  of  the  rays  of  light  coming  to  us,  so  that  the 
direction  in  which  we  have  to  point  a  telescope  in  order  to 
observe  a  star  is  not  the  same  as  if  the  earth  were  at  rest. 

"We  may  illustrate  the  effect  of  aberration  in  the  following 
manner: — A  man  standing  still  in  a  shower  of  rain  when 
the  drops  are  falling  vertically  will,  in  order  to  shield  him- 
self, hold  an  umbrella  right  over  his  head.  But  if  he  proceed 
to  walk  or  run  he  will  find  that  the  drops  seem  to  strike  him 
in  the  face,  so  that  he  has  to  hold  the  umbrella  before  him. 
Also  the  more  he  increases  the  rate  at  which  he  is  moving 
the  greater  will  be  the  deflection  in  the  direction  of  the  rain 
drops.  This  deflection  we  might  call  the  aberration  of  the 
rain. 

Effect  of  Aberration  on  a  Star. 

108.  Let  0  (fig.  61)  be  the  position  of  the  earth ;  draw  OA 
a  tangent  to  the  earth's  orbit  at  0,  and  cut  off  OA  to  represent 
v,  the  velocity  of  the  earth.    Then  let  OS  be  the  direction  of 
a  star  and,  produce  SO  to  B,  so  that   OB  may  represent 
F",  the  velocity  of  light.   Now  in  order  to  be  able  to  consider 
the  question  as  if  the  earth  were  at  rest,  let  us  apply  a  velocity 
equal  and  opposite  to  v  to   both  the  earth   and   to   light. 
This  leaves  the  relative  motion  unaltered.     The  point  0  is 
thus  reduced  to  a  state  of   rest,   while  the  light  may  be 
supposed  to  have  two  velocities  00  and   OB  which  give  a 
resultant  velocity   OD.     The  star  will  therefore  appear  in 
the  direction  OS'  the  production  of  OD  and  the  angle  SOS' 
or  a,  which  measures  the  amount  of  the  displacement,  is  called 
the  aberration  of  the  star. 


CHAP.  VIII.l 


EFFECT   OF    ABERRATION. 


145 


Definitions. 

(1).  The  angle  a  between  the  real  and  apparent  directions 
of  the  star  is  called  the  aberration. 

(2).  The  angle  between  the  real  direction  of  a  star  and 
the  direction  of  the  earth's  motion  is  called  the  earth's  ivay. 
Thus  the  L  BOA  or  the  L  E  is  the  earth's  way. 

From  the  above 
it  is  seen  that  the 
effect  of  aberration  is 
to  displace  a  star  in 
the  direction  of  the 
earth's  motion.  As 
the  direction  of  the 
earth's  motion, 
being  a  tangent  to 
its  orbit,  must  be 
at  right  angles  to 
the  direction  of  the 
sun  ;  therefore,  at 
any  instant  the 
earth  seems  to  be 
moving  to  a  point 
on  the  ecliptic  90° 
behind  the  sun ;  and 
the  displacement  of 
each  star  in  the  heavens,  owing  to  aberration,  takes  place 
along  the  great  circle  joining  its  position  on  the  celestial 
sphere  to  this  point. 

Since  the  sun's  apparent  motion  in  the  ecliptic  is  from 
west  to  east,  and  as  the  longitudes  of  all  heavenly  bodies  are 
measured  from  Aries  in  the  same  direction,  hence  the  point 
on  the  ecliptic  90°  behind  the  sun  is  that  point  whose  longi- 
tute  is  less  than  that  of  the  sun  by  90°.  Thus  if  the  sun's 
longitude  be  120°,  all  stars  will  aberrate  towards  that  point 
of  the  ecliptic  whose  longitude  is  30°. 


FIG.  61. 


146  PRECESSION,  NUTATION,  ABERRATION.          [(3HAP.  VIII, 

Aberration  Varies  as  the  sine  of  the  Earth's  Way. 
109.  We  have,  in  the  triangle  OCD, 


_ 

amCOD~  CD'  V 
sin  a 


or 


.        , 
sin  j3 

.-.     sin  a  =  .ZTsin  ]3  ; 

but  a  being  small,  sin  a  =  a  (in  circular  measure),  and  /3 
may  be  taken  equal  to  the  angle  E  or  the  earth's  way,  as 
they  differ  by  a  very  small  amount  ; 

.*.     «  =  aberration  =  K  sin  E. 

K  is  called  the  coefficient  of  aberration,  and,  when  expressed 
in  circular  measure,  may  be  defined  as  the  ratio  of  the 
velocity  of  the  earth  to  that  of  light. 

If  the  aberration  be  expressed  in  seconds,  we  have 

«»    •     » 

206265  =  V  * 


.-.     0"  =  20"-6  sin  E  nearly. 

Therefore,  the  coefficient  of  aberration  expressed  in  seconds 
is  about  20"-6,  a  more  accurate  value  being  20"*49.  It  is 
evident  that  the  aberration  is  a  maximum  when  the  earth's 
way  =  90°  ; 

.-.     maximum  aberration  =  20"-49  sin  90°  =  20"'49. 

EXAMPLE. 

A  star  in  the  ecliptic  has  a  longitude  of  75°,  obtain  the  change  in  the  position 
of  the  star  owing  to  aberration,  when  the  longitude  of  the  sun  is  135°,  assuming 
the  constant  of  aberration  to  be  20"-49. 

Here  the  angular  distance  of  star  from  sun  =  135°  -  75°  =  60°  ;  .*.  the  earth's 
way  =  30°  since  the  direction  of  earth's  motion  is  at  right  angles  to  direction 

ofsun;  .-.  a  =JST  sin  ^  =  20"  -49  sin  30° 

=  10"-245. 

110.  The  effect  of  aberration  is  to  cause  each  star  to 
appear  to  describe  a  small  ellipse  round  its  true  position  in 


€HAP.  VIII.]  LAW    OF   ABERRATION.  147 

the  course  of  a  year.  This  can  be  shown  in  somewhat  the 
same  way  as  in  the  case  of  annual  parallax.  For  we  may 
regard  the  earth's  orbit  as  being  approximately  a  circle, 
and  that  its  velocity  throughout  the  year  is  uniform.  We 
may  therefore  suppose  each  star  to  move  in  a  circle, 
parallel  to  the  earth's  orbit,  round  its  true  position  as  centre. 
But  when  this  imaginary  circle  is  projected  obliquely  on 
the  surface  of  the  celestial  sphere  it  becomes  an  ellipse  of 
which  the  semi-axis  major  is  parallel  to  the  ecliptic,  and 
equal  to  20"-49  (the  maximum  aberration),  the  semi-axis 
minor  being  20"*49  sin  /,  when  /  is  the  latitude  of  the 
star.  Therefore,  summing  up,  we  have — 

(1)  Each  star  aberrates  towards  a  point  on  the  ecliptic  90° 
behind  the  sun. 

(2)  The  displacement  varies  as  the  sine  of  the  earth's  way. 

(3)  A  star  situated  at  the  pole  of  the  ecliptic  (that  is,  with 
latitude  =  90°)  will,  in  the  course  of  a  year,  appear  to  revolve 
round  its   true  position   in  a  circle  whose  angular  radius   is 
20"*49. 

(4)  A  star  situated  on  the  ecliptic  (that  is,  with  zero  lati- 
tude) will  appear  to  oscillate  through  an  arc  on  the  ecliptic 
of  20" '49  on  either  side  of  its  true  position,  the  total  annual 
displacement  being  40"*9. 

(5)  In  general,  a  star  whose  latitude  is  I  will,  throughout 
the  year,  appear  to  describe  a  small  ellipse  round  its  true  position 
as  centre,  the  semi-axis  major  being  20"'49,  and  parallel  to  the 
ecliptic  and  the  semi-axis  minor  20"*49  sin  I. 

The  student  will  easily  see  that  these  results  differ  con- 
siderably from  those  obtained  in  the  case  of  annual  parallax, 
although  they  have  some  points  of  similarity.  For  the 
annual  parallax  of  a  star  depends  on  its  distance  from  us, 
whereas  the  constant  of  aberration  is  the  same  for  all  stars, 
irrespective  of  their  distances.  Also  in  the  particular  case, 
then  a  star  is  either  in  the  same  part  of  the  celestial  sphere 

L  2 


148  PRECESSION,    NUTATION,    ABERRATION.       [CHAP.  VIII. 

as  the  sun  or  on  the  diametrically  opposite  point,  the  annual 
parallax  is  zero,  but  the  aberration  is  a  maximum.  The 
displacement  due  to  parallax  takes  place  towards  the  sun, 
and  that  due  to  aberration  towards  a  point  on  the  ecliptic 
90°  behind  the  sun. 

111.  The  aberration  of  a  planet  differs  somewhat  from 
that  of  a  star,  being  due  to  two  causes — (1)  That  due  to 
the  velocity  of  the   earth,  and  (2)  to  the  velocity  of  the 
planet.     If  the  planet's  motion  were  equal  to  that  of  the 
earth,  and  in  the  same  direction,  there  would  be  no  aber- 
ration.     In  general,  it  is  easy  to  calculate  the  aberration 
due  to  those  two  causes  separately. 

As  the  velocity  of  the  moon  about  the  earth  is  very 
small  compared  with  the  velocity  of  light,  we  may  regard 
the  aberration  due  to  this  velocity  as  zero.  Neither  is  there 
any  aberration  on  account  of  the  earth's  orbital  motion 
round  the  sun,  for  this  motion  is  shared  in  by  the  moon. 
"We  may,  therefore,  regard  the  moon  as  having  practically 
no  aberration. 

Discovery  of  Aberration. — Bradley  was  first  led  to 
the  discovery  of  aberration  while  attempting  to  find  the 
annual  parallax  of  y  Draconis.  Observing  that  the  latitude 
of  this  star  was  subject  to  small  annual  variations  for  which 
he  could  not  account  by  attributing  them  to  any  known 
cause,  he  was  eventually  led  to  adopt  the  above  explanation. 

112.  Diurnal    Aberration. — Owing    to    the     earth's 
rotation  on  its  axis,  a  point  on  the  equator  turns  through 
25,000  miles  in  23h  56m.     This  is  at  the  rate  of  ^th  of  a 
mile  per  second,  or  g^th  of  the  velocity  of  the  earth  in  its 
orbit.     Any  other  point  on  the  earth  not  on  the  equator  will 
have,  of  course,  a  less  velocity  than  this. 

The  aberration  due  to  this  motion  is  called  diurnal 
aberration.  It  is,  however,  as  we  can  easily  see  by  com- 
paring the  above  velocity  of  rotation  with  that  of  light, 
so  small  as  to  bo  almost  inappreciable. 


(     149     ) 


CHAPTER  IX. 

THE   MOON. 

113.  Next  to  the  sun  the  moon  is  to  us  the  most 
important  of  all  the  heavenly  bodies.  Besides  its  diurnal 
motion  from  east  to  west,  which  is  imparted  to  it  in  common 
with  all  the  other  heavenly  bodies  in  consequence  of  the 
rotation  of  the  earth  on  its  axis,  it  has,  like  the  sun,  a 
motion  among  the  fixed  stars  in  the  opposite  direction, 
making  a  complete  revolution  of  the  heavens  in  about 
27d  7h  43m.  As  the  sun  appears  to  make  a  complete  revo- 
lution of  the  ecliptic  in  one  year,  we  see  that  the  moon's 
motion  among  the  fixed  stars  is  about  thirteen  times  faster 
than  that  of  the  sun..  So  rapid  is  this  motion,  that  its 
change  of  position  with  respect  to  bright  stars  in  its  neigh- 
bourhood can  be  easily  seen,  even  after  as  short  an  interval 
as  two  or  three  hours. 

The  moon's  path,  on  being  mapped  out  on  a  celestial 
globe,  is  found  to  be  represented  by  a  great  circle,  cutting 
the  ecliptic  at  an  angle  of  5°  9',  from  which  it  follows  that, 
like  the  planets,  it  is  always  to  be  found  near  the  ecliptic, 
its  north  or  south  latitude  never  exceeding  5°  9'. 

The  moon's  motion  among  the  fixed  stars  is  due  to  an 
orbital  motion  round  the  earth.  In  fact,  the  moon  is  the 
earth's  satellite.  We  must  not,  however,  suppose  that  its 
orbit  round  the  earth  is  a  circle,  because  the  projection 
of  this  orbit  on  the  celestial  sphere,  on  being  traced  out, 
is  represented  by  a  great  circle.  No,  for  just  as  in  the  case 
of  the  sun,  we  find  that  the  moon's  distance  from  the  earth 
is  not  constant.  We  are  led  to  this  conclusion  by  the  fact 


150  THE  MOON.  [CHAP,  ix, 

that  its  angular  diameter,  on  being  measured  at  different 
times  by  means  of  a  micrometer,  is  found  to  undergo  periodic 
changes,  which  shows  that  its  distance  from  the  earth  must 
be  changing  also,  being  least  when  the  apparent  diameter 
is  greatest.  Its  greatest  angular  diameter  is  33  J' ;  least 
291',  and  the  mean  3iy,  or  a  little  more  than  half  a  degree. 
These  changes  in  the  apparent  angular  diameter  lead  us 
to  the  conclusion — (1)  that  the  moon's  orbit  round  the  earth 
is  approximately  elliptic  with  the  centre  of  the  earth  situated 
in  one  of  the  foci,  (2)  the  radius  vector  joining  the  centres  of 
the  earth  and  moon  sweeps  out  equal  areas  in  equal  times. 

From  this  we  might  infer  that  the  moon's  motion  among 
the  fixed  stars  is  not  uniform.  In  fact,  it  varies  from  a 
maximum  of  33'  40"  per  hour  to  a  minimum  of  27',  its 
mean  hourly  velocity  being  32'  56".  So  that  we  may  say 
that  the  moon  in  its  motion  among  the  fixed  stars  moves 
through  an  arc  equal  to  its  own  diameter  in  one  hour. 

The  mean  distance  of  the  moon  from  the  earth  is 
238,000  miles,  or  about  60  times  the  earth's  radius.  As 
this  distance  is  much  less  than  the  radius  of  the  sun,  which 
is  110  times  the  radius  of  the  earth  (Art.  44),  we  see  that 
if  the  sun  were  placed  with  its  centre  at  the  centre  of  the 
earth  its  mass  would  extend  considerably  beyond  the  moon, 
a  consideration  which  will  perhaps  enable  the  mind  to  form 
some  idea  of  the  magnitude  of  the  body  which  forms  the 
centre  of  our  system. 

The  Moon's  Phases 

114.  One  of  the  most  interesting  phenomena  to  be  seen 
in  the  heavens  is  the  series  of  changes  which  the  visible 
portion  of  the  moon's  illuminated  surface  presents  during 
its  orbital  motion  about  the  earth.  These  appearances  are 
called  its  phases.  They  prove  that  the  moon  is  an  opaque 
spherical  body  deriving  its  light  from  the  sun.  As  only  one 


CHAP.  IX.]  MOON'S  PHASES.  151 

hemisphere  of  the  moon  can  be  illuminated  at  once,  viz. 
that  half  which  is  turned  towards  the  sun,  an  observer 
will  therefore  see  a  variable  amount  of  this  bright  surface 
depending  on  the  relative  positions  of  the  sun,  moon,  and 
earth. 

Let  ACMD  (fig.  62)  represent  the  orbit  of  the  moon, 
E  the  earth,  and  S  the  direction  of  the  sun.  In  the  eight 
positions  of  the  moon,  which  we  have  here  depicted,  the 
line  mn,  which  is  perpendicular  to  the  direction  of  the  sun, 


O 


FIG  62. 

separates  the  illuminated  half  of  the  moon  from  the  unillu- 
minated  half,  and  all  the  positions  of  mn  are  drawn  as  if 
parallel  to  one  another,  the  sun  being  so  far  distant.  The 
line  ab  may  be  taken  as  separating  the  half  of  the  moon 
which  is  turned  towards  the  observer  from  that  which  is 
turned  away  from  him. 

When  the  moon  is  in  conjunction  at  A,  its  dark  hemi- 
sphere is  turned  towards  the  earth,  and  no  portion  is  visible 
to  the  observer.  It  is  then  said  to  be  new  moon. 


152  THE  MOON.  [CHAP.  ix. 

Some  four  or  five  days  afterwards,  when  the  moon  is  at 
By  the  observer  will  see  a  small  portion  of  the  illuminated 
surface  which  will  appear  as  a  thin  crescent  in  the  sky, 
seen  in  the  west  after  sunset. 

When  the  moon  is  at  C,  90°  from  the  sun ;  that  is,  in 
quadrature,  it  will  appear  in  the  sky  as  a  bright  semicircle. 
This  is  said  to  be  first  quarter,  and  the  moon  is  then  said  to 
be  dichotomized. 

At  D  it  is  gibbous,  and  when  in  opposition  at  M,  which 
occurs  at  about  15  days  after  conjunction,  the  whole  of  the 
illuminated  hemisphere  is  turned  towards  the  observer.  The 
moon  will  then  present  a  complete  circular  disc  in  the  sky. 
This  is  said  to  be  full  moon. 

After  full  moon,  these  phases  are  repeated  in  reverse 
order,  the  moon  being  again  in  quadrature  at  G,  which 
is  called  third  quarter,  and  finally,  conjunction  is  once  more 
reached  at  A. 

When  in  conjunction  and  opposition,  the  moon  is  said 
to  be  in  syzygy.  Its  elongation  from  the  sun  is  then  0°  and 
180°,  respectively.  When  in  quadrature  at  first  quarter  its 
elongation  is  90°,  and  at  third  quarter  270°. 

Definitions. 

(1).  The  time  taken  by  the  moon  to  make  a  complete 
revolution  with  reference  to  the  fixed  stars  is  called  its  periodic 
time  or  sidereal  period.  This  period  is  27d  7h  43m. 

(2),  The  interval  between  two  successive  conjunctions  or 
oppositions,  or,  in  other  words,  the  time  taken  to  make  a 
complete  revolution  with  reference  to  the  sun  is  called  ,the 
synodic  period  or  a  lunation.  This  period  is  29£  days,  or, 
more  accurately,  29-5305887  days. 

It  is  obvious  that  if  the  sun  had  no  apparent  motion  in 
the  ecliptic,  the  synodic  and  sidereal  periods  would  be  exactly 
the  same,  so  that  the  full  moons  would  follow  one  another 
at  intervals  of  27d  7h,  instead  of  291  days.  But  while  the 


CHAP.  IX.]  SYNODIC  PERIOD   DETERMINED.  153 

moon  is  making  a  complete  revolution  round  the  earth, 
which  it  does  in  27d  7h,  the  sun  moves  through  an  arc  of 
about  27°  on  the  ecliptic  in  the  same  direction  (roughly  at 
the  rate  of  1°  daily),  so  that  it  takes  the  moon  an  additional 
two  days  to  arrive  at  the  same  position  relative  to  the  sun 
and  earth  as  when  it  started.  In  the  above  diagram  illus- 
trating the  phases  of  the  moon  we  have,  for  the  sake  of 
simplicity  of  explanation,  supposed  the  sun  and  earth  fixed, 
and  that  the  moon  moves  with  its  relative  velocity  with 
respect  to  the  sun,  completing  the  revolution  in  29^-  days. 

To  determine  the  Moon's  Synodic  Period. 

115.  We  know  that  when  an  eclipse  of  the  moon  takes 
place,  the  moon  must  be  in  opposition.     Therefore,  if  we 
observe  the  exact  interval  of  time  that  elapses  between  the 
middle  of  two  eclipses,  and  divide  by  the  number  of  luna- 
tions between  them  we  get  the  length  of  a  single  lunation 
or  synodic  period. 

The  mean  length  of  a  lunation  can  be  calculated  very 
accurately  from  the  records  of  ancient  eclipses.  The  earliest 
observations  of  eclipses  of  which  there  is  an  accurate  account 
are  those  taken  at  Babylon  in  the  years  720  and  719  B.C. 
The  number  of  lunations  between  one  of  these  eclipses  and 
an  eclipse  at  the  present  day  being  known,  we  are  able 
to  calculate  the  mean  value  of  a  lunation  over  a  very  long 
period  of  time. 

To  find  the  Moon's  Sidereal  Period. 

116.  Knowing  the  moon's  synodic   period,  we  are  able 
to   calculate   its   sidereal   period,   or   periodic   time,  in  the 
same  way  as  in  the  case  of  a  planet  (Art.  67).     In  fig.   63 
E  represents  the   earth,   the   inner   circle   being  the  orbit 
of  the  moon,  and  the  outer  circle  the  apparent  orbit  of  the 
sun  about  the   earth.     A  and  B  are  the  positions  of  the 


154  THE  MOON.  [CHAP.  ix. 

moon  and  sun  at  conjunction,  and  A'  and  B'  their  positions 
one  day  after  conjunction. 
Let 

S  =  Period  of  sun's  motion  about  earth  =  365^  days. 

P  =  Moon's  periodic  time  or  sidereal  period. 

L  =  Interval  between  two  conjunctions  =  2S-J-J-  days. 


but 


FIG.  63. 


'•     -p-  =  L  described  by  moon  in  1  day  =  L  AEA', 

Q/2A 

—  ~-  =  L  described  by  sun  in  1  day  =  L 

360     360 

""     ~P  ---  8~  =  L 


day  = 


=  L  gained  by  moon  in  1  day,  also: 


360_360  _36Q 
P  '  ~S~     ~L 


1 

or    —  - 


1 


P     365-25     29-5 


CHAP.  IX.]  METONIC   CYCLE.  155 

therefore,  solving  for  P,  we  find  the  periodic  time  to  be 
27d  7h  nearly. 

A  more  accurate  value  for  the  periodic  time  is  27d  7h 
43m  11s,  while  that  of  a  lunation  is  29-5305887  days. 

Metonic  Cycle. 

117.  Meton   first   discovered,   B.C.   433,   that   19    years 
expressed  in  days  is  an  almost  exact  multiple  of  a  lunation, 
for  365-25  x  19  =  6939-75  and  29-5305887  x  235  =  6939-688. 
So  that  in  every  19  years  there  are  almost  exactly  235  luna- 
tions.    Therefore,  at  the  end  of  every  19  years  the  sun  and 
moon,  returning  to  the  same  positions  with  respect  to  the 
fixed  stars,  all  the  phases  of  the  moon  will  occur  again  on 
the  same  days  of  the  month  as  for  the  previous  19  years,  the 
only  difference  being  that  they  will  occur  about  one  hour 
sooner.     This  is  called  the  Metonic  Cycle.     The  discovery  of 
the   Metonic  cycle  was   of  considerable   importance,   as   it 
afforded  a  ready  method  of  predicting  the  dates  of  the  full 
moons,  etc.,  without  the  trouble  of  calculation.      It  has  been 
much  used  in  order  to  find  the  date  on  which  Easter  should 
fall  in  a  given  year,  because  this  festival  occurs  on  the  Sunday 
following  the  first   full  moon  after  the  21st  March.     For 
this  reason,  the  nineteen  numbers,  from  1  to  19,  are  called 
the  golden  numbers.     The  golden  number,  or  the  number  in 
the  Metonic  Cycle,  for  any  year,  is  the  remainder  got  after 
dividing  the  year  increased  by  unity  by  19.     Thus   the 
golden   number  for  1901  is  the  remainder  when  1902  is 
divided  by  19,  i.  e.  2.     Where  zero  is  the  remainder,  then  19 
is  the  golden  number. 

Apparent  area  of  illuminated  Surface  of  Moon. 

118.  It  can  be  shown  in  exactly  the  same  way  as  in  the 
case  of  a  planet  (Art.  62)  that  the  apparent  area  of  the 
bright  portion  turned  towards  the  earth  is  proportional  to 


156  THE  MOON.  [CHAP.  ix. 

the  versed  sine  of  the  exterior  angle  subtended  at  the  moon 
by  the  earth  and  sun.  Thus,  if  (fig.  64)  M  represent  the 
moon,  E  the  earth,  and  8  the  sun,  the  external  angle  at  the 
moon  is  the  angle  a  ;  therefore  apparent  area  varies  as  versin  a ; 
but  (Euclid,  I.  32),  a  =  j3  +  0.  Therefore  the  angle  a  is 
very  nearly  equal  to  j3,  for  the  moon,  being  so  near  the  earth, 
6  is  always  a  very  small  angle,  being  never  more  than  10'. 
Therefore  the  apparent  area  varies  approximately  as  versin  /3 
where  |3  is  the  angle  of  elongation  of  the  moon  from  the  sun. 
Of  course  this  approximation  is  not  true  in  the  case  of  a 
planet,  for  its  distance  from  the  earth  being  so  very  much 
greater  than  that  of  the  moon,  we  could  by  no  means  neglect 
the  angle  0. 


FIG.  64. 

119.  Earth-shine. — It  is  evident  that  if  the  earth  were 
seen  from  the  moon  it  would  appear  to  pass  through  the 
same  phases  as  the  moon  does  when  observed  from  the  earth, 
but  in  inverse  order.  During  new  moon  the  earth,  as  seeu 
from  the  moon,  would  appear  full.  When  the  moon  appears 
as  a  crescent  the  earth  would  appear  gibbous,  and  vice  versa. 
This  accounts  for  the  phenomenon  which  doubtless  everyone 
has  observed,  that  when  the  moon  appears  as  a  thin  crescent 
in  the  sky  the  remainder  of  its  surface  can  be  seen  shining 
with  a  dull  grey  light,  caused  by  the  earth-shine  on  the  moon, 
which  is  reflected  back  again  to  the  earth. 


CHAP.  TX.]  MOON'S    ROTATION.  157 


To  find  the  Sun's  distance  by  observing  when  the  Moon  is 
Dichotomized. 

120.  In  Chapter  VII.  we  described  the  different  methods 
by  which  the  sun's  distance  can  be  calculated.  There  is 
another  method  however,  which,  although  not  susceptible  of 
the  same  degree  of  accuracy  as  those  employed  in  modern 
times,  is  of  great  historical  interest,  as  it  was  used  by  Aris- 
tarchus  at  Alexandria  about  280  B.C.,  being  the  first  attempt 
at  determining  the  sun's  distance. 

The  angle  of  elongation  j3  (fig.  64)  of  the  moon  from  the 
sun  is  observed  when  the  moon  is  dichotomized,  or,  in  other 
words,  when  the  L  SMJE=  90°. 


AT  Q 

Now,  COS0  =  — , 

and  j3  being  known,  the  ratio  of  the  moon's  distance  to  the 
sun's  is  known,  from  which,  knowing  that  of  the  moon,  the 
distance  of  the  sun  is  determined. 

It  is  not  possible  to  obtain  accurate  results  by  this  method, 
as,  owing  to  inequalities  in  the  moon's  surface,  the  line  which 
separates  the  bright  from  the  dark  portion  is,  when  seen 
through  a  telescope,  very  uneven,  so  that  the  observer  is 
unable  to  tell  the  exact  instant  when  the  moon  is  dichoto- 
mized. Aristarchus  deduced  by  this  method  that  the  sun 
was  19  times  more  distant  than  the  moon,  instead  of  400 
times,  which  modern  observations  give  us. 

The  Moon  rotates  round  an  Axis. 

121 .  It  is  a  remarkable  circumstance  in  connexion  with  the' 
moon  that  it  always  turns  nearly  the  same  face  to  the  observer. 
The  mountains  and  other  markings  which  are  to  be  seen  on  its 
surface  are  always  to  be  found  nearly  in  the  same  position 
with  respect  to  the  circumference  of  the  moon's  disc,  and  also 


158  THE  MOON.  [CHAP.  ix. 

relative  to  the  plane  of  its  orbit.     From  this  circumstance  we 
are  led  to  conclude  that — • 

(1)  The  moon   revolves  round  an  axis  which  is  nearly 
perpendicular  to  the  plane  of  its  orbit, 

(2)  The  period  of  its  rotation  round  its  axis  must  be  equal 
to  the  time  of  completing  a  revolution  round  the  earth,  viz. 
27d  7h.     On  first  thoughts  it  might  appear  to  the  reader 
as  if  the  fact  that  the  moon  keeps  the  same  face  turned 
towards  the  earth  proves  that  it  has  no  rotation.  The  follow- 
ing illustration  will  serve  to  show  how  erroneous  is  such  a 
conclusion : — Let  the  reader  place  a  lamp  or  other  body  in 
the  middle  of  a  room,  and  let  him  proceed  to  walk  round  it 
in  a  circle  so  as  to  keep  his  face  turned  towards  it  all  the 
time.     Now,  let  us  suppose  that  at  first  his  face  is  turned 
towards  the  north,  and  he  will  find,  while  he  is  completing  a 
circuit,  that  he  faces  in  turn  towards  all  the  points  of  the 
compass.     He  will  be  looking  towards  the  south  when  he  has 
moved  through  a  semicircle,  and  will  again  face  the  north 
when  he  arrives  at  the  point  from  which  he  started.     In 
other  words,  in  order  to  keep  his   face   turned  constantly 
towards  the  lamp  he  will  have  to  rotate  his  body  through 
360°  for  every  circuit  he  makes.     So  it  is  in  the  case  of  the 
moon's  revolution  round  the  earth. 

Moon's  Librations — Libration  in  Latitude. 

122.  The  axis  of  rotation  of  the  moon  is  not  quite  per- 
pendicular to  the  plane  of  its  orbit,  being  inclined  to  it  at  an 
angle  of  83J°,  or  about  6^°  to  the  perpendicular.  Therefore 
while  the  moon  revolves  about  the  earth  its  north  and  south 
poles  are  alternately  turned  slightly  towards  or  from  the 
observer.  At  one  part  of  its  orbit  we  see  about  6JC  beyond 
the  north  pole,  and  at  another  time  about  6|°  beyond 
the  south  pole.  This  phenomenon  is  called  libration  in 
latitude. 


CHAP.  IX.]  THE    MOON'S    ITERATIONS.  159 

Libration  in  Longitude. 

We  have  seen  that  the  period  of  the  moon's  rotation  on 
its  axis  is  equal  to  the  time  taken  to  go  round  the  earth. 
But  its  motion  round  the  earth  is  not  uniform,  as,  owing  to 
the  elliptic  form  of  its  orbit,  its  distance  from  the  earth  is 
not  constant.  On  the  other  hand  its  rotation  on  its  axis  is 
perfectly  uniform.  The  consequence  is,  that  although  the 
two  periods  of  making  a  complete  revolution  are  the  same 
for  each,  still  at  one  time  we  are  ahle  to  see  a  little  more  of 
the  eastern  side,  and  at  another  time  a  little  more  of  the 
western  side.  This  is  called  libration  in  longitude.  Its  maxi- 
mum amount  is  7°  45'. 

Diurnal  Libration. 

There  is  also  a  diurnal  libration  which  is  really  due  to 
parallax.  For,  from  the  time  the  moon  rises  until  it  sets,  the 
observer,  on  account  of  the  rotation  of  the  earth,  has  changed 
his  point  of  observation,  and  therefore  he  does  not  in  each 
case  see  exactly  the  same  face.  When  the  moon  is  rising  in 
the  east  he  sees  a  little  more  of  its  western  side,  and  when 
setting  in  the  west,  a  little  more  of  its  eastern  side,  than  when 
it  is  high  up  in  the  sky  crossing  the  meridian. 

The  total  effect  of  these  librations  is  such  that  we  are  at 
different  times  able  to  see  a  total  of  about  59  per  cent,  of  the 
moon's  surface  instead  of  about  50  per  cent. 

Path  of  the  Moon  round  the  Sun. 

123.  We  have  seen  that  the  moon's  orbit  relative  to  the 
earth  is  an  ellipse,  but,  as  it  also  follows  the  earth  in  its 
motion  round  the  sun,  we  see  that  the  path  of  our  satellite 
round  the  sun  is  due  to  a  combination  of  two  motions,  a 
monthly  motion  about  the  earth,  and  a  yearly  motion  about 
the  sun.  If  we  neglect  the  small  angle  at  which  its  orbit 
cuts  the  plane  of  the  ecliptic,  and  assume  them  both  in  the 
same  plane,  the  moon's  path  may  be  represented  by  the 


160  THE  MOON.  [CHAP.  ix. 

dotted  line  in  the  adjoining  figure,  going  alternately  inside 
and  outside  the  orbit  of  the  earth  AEE',  and  crossing  it  about 
25  times  in  the  course  of  the  year.  M  represents  the  moon, 
and  E  the  earth  at  new  moon,  while  H'  and  E'  would  be 
their  positions  at  the  next  full  moon  after  an  interval  of  about 
a  fortnight.  It  is  to  be  observed  that  this  path  of  the  moon 
is  always  concave  to  the  sun. 


.  65. 

More  Moonlight  in  Winter  than  in  Summer. 
The  moon,  when  full,  being  in  opposition,  must  be  at 
almost  the  diametrically  opposite  point  of  the  celestial  sphere 
to  that  in  which  the  sun  is  situated.  Hence  at  full  moon  at 
midsummer  the  sun's  declination  being  north  the  moon  must 
have  an  equal  southern  declination,  and  therefore  remains 
but  a  short  time  above  the  horizon  (Art.  20).  Again,  at  full 
moon  during  midwinter  the  conditions  are  reversed,  the  sun's 
decimation  is  south,  and  the  moon's  north ;  hence  we  have  the 
moon  a  long  time  above  the  horizon.  This  happens  just 
when  the  days  are  shortest  and  we  are  most  in  need  of  light. 

Moon's  Retardation. 

124.  The  moon  moves  from  west  to  east  with  reference  to 
the  sun  through  360°  every  29J  days,  or  about  12J°  daily. 
Therefore  its  time  of  rising  will  be  later  and  later  each  night 
by  an  interval  whose  mean  value  is  about  50  minutes.*  This 
retardation,  as  it  is  called,  of  moonrise  is  not  by  any  means 
uniform  throughout  the  year ;  it  may  be  as  great  as  1  hour 
16  minutes  or  as  small  as  17  minutes. 

*  Since  15°  correspond  to  1  hour,  12£°  are  equivalent  to  50  minutes. 


PHAP.   IX.]  HARVEST    MOON.  161 

Harvest  Moon. 

125.  At  the  full  moon  nearest  the  autumnal  equinox  it 
:s  found  that  the  retardation  in  the  time  at  which  the  moon 
rises  is  less  than  at  any  other  full  moon  throughout  the  year 
Therefore  for  several  nights  in  succession  the  moon  will  rise 
yery  shortly  after  sunset,  rising  on  the  night  of  full  moon,  as 
it  always  does,  at  sunset.  As  this  happens  when  the  farmers 
are  getting  in  their  crops,  thus  enabling  them  to  prolong 
their  work  into  the  night,  it  is  called  the  Harvest  Moon. 


FIG.  65A. 


In  order  to  explain  this  phenomenon  more  clearly  we 
shall  suppose  that  the  moon's  path  is  along  the  ecliptic  in- 
stead of  being  inclined  to  it  at  a  small  angle,  and  that  it 
moves  uniformly  in  the  ecliptic  at  the  rate  of  13j°  daily 
('360°  in  27d  7h),  and  we  may  remark  that,  owing  to  the  ap- 
parent diurnal  revolution  of  the  heavenly  bodies  the  angle  at 
which  the  ecliptic  cuts  the  horizon  is  continually  changing, 
its  greatest  and  least  values  being  colat  +  23°  28'  and  colat 
-  23°  28'  respectively.  The  reason  of  this  is  that  the  pole 
of  the  ecliptic,  which,  in  its  diurnal  motion,  describes  a 
small  circle  mn  (fig.  65A)  of  angular  radius  23°  28'  round 

M 


162  THE  MOON.  [CHAP.  ix. 

the  celestial  pole,  is  closest  to  the  zenith  at  m  when 
Zm  =  ZP-  Pm  =  colat  -  23°  28',  and  at  its  greatest  zenith 
distance  at  n  when  Zn  =  colat  +  23°  28'.  Hence  the  angle 
between  the  ecliptic  and  horizon  (being  equal  to  the  angular 
distance  between  their  poles)  must  vary  between  the  same 
limits,  being  least  when  r  is  rising  at  the  east  point  X  when 
the  ecliptic  KC  passes  between  the  horizon  and  equator,  the 
order  being  horizon,  ecliptic,  equator ;  and  greatest  when  ^  is 
at  X  when  the  ecliptic  takes  the  position  jBT'O',  the  order 
being  horizon,  equator,  ecliptic. 

During  the  full  moon  nearest  the  autumnal  equinox  the 
sun  is  in  Libra,  and  the  moon,  being  in  opposition,  is  in  Aries, 
crossing  from  the  south  to  the  north  side  of  the  equator ;  the 
moon  will  therefore  rise  at  X,  when  the  ecliptic  KC  is  at  its 
smallest  inclination  (colat  -  23°  28')  to  the  horizon.  But 
after  an  interval  of  23h  56m  when,  owing  to  the  diurnal 
revolution  of  the  celestial  sphere,  the  point  X  returns  to  the 
same  position  as  on  the  previous  night,  the  moon  will  have 
moved  about  13°.  Hence,  if  XM  be  cut  off  on  the  ecliptic 
equal  to  the  moon's  daily  rate,  and  through  M  an  arc  of  a 
small  circle  ML  be  drawn  parallel  to  the  equator,  the  moon, 
on  the  night  following  full  moon,  will  rise  at  L,  the  amount 
of  retardation  being  measured  by  the  arc  of  the  small  circle 
LM  or  by  the  angle  LPM.  This  retardation,  expressed  in 
time,  is  found  to  be  in  our  latitudes  only  18 J  minutes,  or 
14j  minutes  with  reference  to  the  sun  (allowing  for  the  sun's 
daily  retardation  of  4  minutes).  This  phenomenon  of  course 
occurs  each  time  the  moon  is  in  Aries  or,  in  other  words,  every 
month ;  but  it  is  only  during  the  harvest  that  the  moon  is  in 
Aries  and  full  at  the  same  time. 

Similarly  we  might  show  that  the  reverse  occurs  when  the 
sun  is  in  Aries  and  the  moon  in  Libra,  the  daily  retardation 
being  then  a  maximum.  For  when  the  moon  is  rising,  the 
ecliptic  takes  the  position  K'C',  cutting  the  horizon  at  the 
greatest  angle  possible  (colat  +  23°  28') :  if  XM'  be  now  cut 


CHAP.  IX.]  CHANGES  IN  THE  MOON's  NODES.  163 

off  equal  to  the  moon's  daily  rate  of  motion,  and  a  parallel 
3fL'  be  drawn  to  the  equator,  the  moon,  on  the  following 
night,  will  rise  at  L',  the  retardation  being  measured  by  the 
arc  of  the  small  circle  M' L'  or  by  the  angle  M'PL'  (the  arc 
PM  when  produced  passing  through  M.').  This,  expressed  in 
time,  corresponds  to  a  retardation  of  about  lh  10m  with  refe- 
rence to  the  fixed  stars  or  lh  6m  with  reference  to  the  sun. 
The  retardation  is  thus  a  maximum  each  month  when  the 
moon  is  in  Libra,  and  therefore  is  a  maximum  during  the 
full  moon  nearest  the  vernal  equinox.  It  will,  however,  be  a 
minimum  for  observers  in  the  southern  hemisphere,  and  to 
them  the  full  moon  at  this  period  is  a  harvest  moon. 

At  the  arctic  circle  there  is  even  an  acceleration  in  the 
time  at  which  the  moon  rises  on  successive  nights  when  the 
moon  is  in  Aries.  For,  when  Aries  is  rising,  the  ecliptic 
actually  coincides  with  the  horizon  (Ex.  5,  page  32),  and  the 
distance  ML  (fig.  65A)  vanishes,  and  therefore  the  interval 
between  two  successive  risings  is  only  23h  56m.  So  that,  as 
measured  by  solar  time,  at  the  arctic  circle,  the  moon  after 
passing  through  Aries  actually  rises  four  minutes  earlier  than 
she  did  on  the  previous  night. 

During  the  October  full  moon  the  same  phenomena  occur, 
but  in  a  less  marked  degree.  This  moon  is  called  the 
Hunter's  Moon. 

Revolution  of  the  Moon's  Nodes. 

126.  The  moon's  nodes  are  not  fixed  points,  but  have  a 
retrograde  motion  along  the  ecliptic  at  the  rate  of  about  19° 
each  year,  completing  a  revolution  in  about  18  J  years.  This 
backward  motion  is  similar  to  that  of  the  equinoctial  points 
v  and  •£=,  but  is  very  much  more  rapid,  as  the  period  for  the 
precession  of  the  equinoxes  is  about  26,000  years  (Art.  103). 
The  moon's  motion  is  therefore  very  complicated,  moving,  as 
it  appears  to  do,  in  a  circle  which  is  inclined  to  the  ecliptic 
at  an  angle  of  5°,  at  the  rate  of  rather  more  than  half  a 

M  2 


164  THE  MOON.  [CHAP.  ix. 

degree  each  hour,  while  the  plane  of  this  circle  is  carried 
backwards  on  the  ecliptic  at  the  rate  of  19°  each  year,  or 
about  8"  an  hour. 

127.  Synodic  Revolution  of  the  Moon's  Modes. — 

We  have  just  seen  that  the  sidereal  period  of  the  revolution 
of  the  moon's  nodes  is  18f  years.  The  synodic  period  of 
revolution,  i.e.  the  time  taken  to  return  from  any  position  to 
the  same  position  again  with  respect  to  the  sun  and  earth, 
can  now  be  calculated  in  the  same  way  as  that  of  a  planet, 

f°r~  360° 

=  arc  traversed  by  sun  in  1  day, 


365-25 

360° 

and      r-^r  —  —  -r—  r=  =  arc  traversed  by  node  in  1  day; 
18f  x  <365'25 

360°  360°  360°  , 

'  •     365-25  +  ^      =  ~       (see  Art  67)' 


where  T  represents  the  synodic  period.  The  plus  sign  is 
taken  on  the  left-hand  side  of  the  equation,  as  the  relative 
velocity  of  the  sun  and  node  is  the  sum  of  their  angular 
velocities,  the  motion  of  the  node  being  retrograde. 

On  solving  the  above  equation,  T  is  found  to  be  346'62 
days,  which  is  the  synodic  period. 

The  line  of  apsides  of  the  moon's  orbit  is  not  fixed,  but, 
like  that  of  the  earth's  orbit,  it  has  a  slow  progressive  motion, 
making  a  complete  revolution  of  the  moon's  orbit  in  about 
nine  years,  the  period  in  the  case  of  the  earth  being  about 
108,000  years. 

To  find  the  Height  of  a  Lunar  Mountain. 

128.  It  has  been  known,  from  the  time  of  Galileo, 
that  the  surface  of  the  moon  is  covered  with  mountains. 
Some  of  these  mountains  have  been  calculated  to  rise  to 
heights  of  four  or  five  miles  above  the  surface  of  the  sur- 
rounding plains,  which  shows  that,  considering  the  smaller 
size  of  the  moon,  its  mountains  are  comparatively  much 


CHAP.  IX.]  HEIGHT  OF  LUNAR  MOUNTAIN.  165 

more  lofty  than  those  of  the  earth.  When  the  sun  shines 
obliquely  on  the  mountains,  they  cast  long  shadows  over 
the  surrounding  plains  on  the  side  remote  from  the  sun 
in  exactly  the  same  way  as  we  are  familiar  with  on  the 
earth.  Also,  wherever  there  is  a  high  mountain,  its  top- 
most peak  catches  the  first  rays  of  the  rising  and  the  last 
of  the  setting  sun,  when  all  the  surrounding  parts  are  still 
in  complete  darkness.  Small  points  of  light  are  for  this 
reason  sometimes  seen  on  the  dark  portion  of  the  moon's 
disc,  separated  by  a  measurable  distance  from  the  line  of 
separation  of  light  and  darkness. 

129.  First  Method. — The  method  employed  by  Messrs. 
Beer  and  Madler  in  1837  for  finding  the  height  of  a  lunar 
mountain  consists  in  measuring,  by  means  of  a  micrometer, 
the  length  of  the  shadow  cast  by  the  mountain  when  illu- 
minated by  the  sun's  rays.      By  comparing  this  angular 
measurement  with   the    angle    subtended    by    the    moon's 
diameter,  the  length  of  the  shadow  in  miles  can  be  found 
(for  the  moon's  diameter  in  miles  is  known) ;  from  which, 
knowing  the  inclination  of  the  sun's  rays,  the  height  of  the 
mountain  can  be  determined,  just  as  the  height  of  a  tower 
on  the  earth  can  be  found  by  knowing  the   length  of  its 
shadow   and  the   altitude   of  the   sun.      In   applying  this 
method,  allowance  must  be  made  for  the  effect  of  foreshort- 
ening, as  the  shadow  being  generally  viewed  obliquely,  the 
micrometer  measures    merely   the   projection   of  its   actual 
length  on  a  plane  perpendicular  to  the  line  of  vision. 

130.  Second  Method. — This  method  consists  in  mea- 
suring, by  means  of  a   micrometer,  the   angular  distance 
between  the  bright  summit  of  the  mountain-top  appearing 
on   the   dark   portion   of  the  moon's  disc  and  the  line  of 
separation   of   light   and   darkness.      This  measurement   is 
made  in  a  direction  perpendicular  to  the  line  joining  the 
extremities  of  the  horns,  and  therefore  parallel  to  the  plane 
of  the  moon,  earth,  and  sun. 


166 


THE    MOON. 


[CHAP,  ix* 


Let  AB  represent  the  line  of  separation  of  light  and 
darkness  on  the  moon,  and  P  the  top  of  a  mountain  when 
just  illuminated  by  the  ray  PBS,  the  line  PS  being  per- 
pendicular to  AB,  and  touching  the  moon's  surface  at  B. 
Let  E  be  the  earth,  and  ES'  the  direction  of  the  sun,  as 


Direction   of  Sun, 
FIG.  66. 

seen  from  E,  which  may  be  taken  as  parallel  to  PS,  on 
account  of  the  sun's  great  distance.  The  radius  of  the 
moon  is  denoted  by  r,  and  the  height  PC  of.  the  mountain 
by  h;  then  by  Euclid  (m.  36), 


or 

that  is, 


(2r  +  h) 
2rh  +  h* 


(see  fig.  66)  ; 


But  h  being  very  small  compared  with  r,  its  square  may 
be  neglected  ; 

/2 

.-.     2rh  =  tz  or  7*  =  —  • 

AT 

Now  the  distance  t  is  not  measured  directly  ;  what  is 
actually  measured  being  the   projection    of   t   on   a    plane 


CHAP.  IX.]  PHYSICAL  STATE  OF  THE  MOON.  167 

perpendicular  to  the  line   of  sight,  viz.   the  perpendicular 
8  let  fall  from  P  on  BE  (fig.  66). 

g 
But    -  =  sin  0  =  sin  0  (by  parallel  lines)  ; 


But  the  angle  0  is  known,  being  the  angle  of  elongation 
of  the  moon  from  the  sun  as  seen  from  the  earth  ;  therefore, 
t  is  known.  Substituting  this  value  of  t,  we  have  — 

S2 

Height  of  mountain  h  =  ?r  —  —  r-  • 
2r  sm2^ 

131.  Lunar  Craters.  —  Perhaps  the  most  striking  ob- 
jects to  be   seen  in  lunar  landscapes  are  what  are  to  all 
appearances  enormous  craters  of  what  were  once  volcanoes, 
but  which  are  now  probably   quite  extinct.     The  typical 
lunar  crater  consists  of  an  immense  circular  plain  surrounded 
by  a  high  wall  or  rampart.      In  the  centre  of  the  plain  there 
generally  rises  a  mountain,  or  sometimes  more  than  one. 
Among  the  most  characteristic  of  these  craters  are  Tycho, 
having   a   diameter   of  fifty-four   miles,    and   Archimedes, 
whose  diameter  is  sixty  miles.     Another  immense  crater  is 
Schickard,  with  a  diameter  of  over  130  miles,  and  a  sur- 
rounding wall,  which,  in  some  parts,  attains  a  height  of 
10,000  feet.     It  has  been  pointed  out  that   an   observer 
situated  in  the   centre   of  this  walled   space   would   think 
himself  in  the  midst  of  a  boundless  desert,  for,  on  account 
of  the  curvature  of  the  moon's  surface,  the  summit  of  the 
lofty   surrounding  wall   would   be  altogether  beneath   his 
horizon. 

132.  Lunar  Atmosphere.  —  All  observers  of  the  moon 
have  come  to  the  conclusion  that  it  either  possesses  no  atmo- 
sphere at  all,  or,  if  any  such  gaseous  covering  exist,  that  it 
is  of  very  extreme  tenuity  indeed.     No  change  is  observed 
in  the  intensity  of  the  light  from  a  fixed  star  as  it  approaches 


168  THE  MOON.  [CHAP,  ix 

the  dark  edge  of  the  moon,  such  as  there  would  he  were 
there  any  appreciable  thickness  of  atmosphere  for  its  rays 
to  penetrate.  Also,  when  the  moon  passes  hetween  the 
observer  and  a  fixed  star,  the  observed  time  during  which 
the  occultation  of  the  star  lasts  is  found  not  to  be  less  than 
the  calculated  time,  as  would  be  the  case  if  the  moon  had 
an  atmosphere  of  any  considerable  density;  for  the  star 
would  still  be  visible  for  some  time  after  being  actually 
covered  by  the  moon,  owing  to  its  rays  being  refracted  in 
their  passage  through  the  lunar  atmosphere,  if  such  existed, 
just  as,  owing  to  refraction  by  the  earth's  atmosphere,  the 
sun  remains  visible  to  us  for  some  time  after  he  has  sunk 
below  our  horizon. 

In  addition  to  having  no  atmosphere,  astronomers  have 
not  been  able  to  detect  water  in  any  form  on  the  moon's 
surface,  which  renders  the  existence  of  life,  such  as  is  known 
to  us,  altogether  impossible. 


(    169    ) 


CHAPTER  X. 

ECLIPSES. 

133.  Eclipses  are  of  two  kinds,  lunar  and  solar. 
Lunar  Eclipses. — A  lunar  eclipse   is   caused  by  the 

passage  of  the  moon  through  the  shadow  of  the  earth. 
This  can  only  happen  when  the  earth  is  between  the  sun 
and  moon,  or,  in  other  words,  when  the  moon  is  in  oppo- 
sition. 

If  the  plane  of  the  moon's  orbit  coincided  with  the  plane 
of  the  ecliptic  instead  of  being  inclined  to  it  at  an  angle 
of  about  5°,  we  should  have  an  eclipse  of  the  moon  at 
every  opposition.  However,  on  account  of  the  above  angle 
of  inclination  of  its  orbit,  it  generally  happens  that  the 
moon,  when  in  opposition,  is  either  so  far  above  or  below 
the  plane  of  the  ecliptic  that  it  fails  to  pass  through  the 
shadow  of  the  earth.  So  we  see  that,  in  order  that  an 
eclipse  may  take  place,  the  moon  must  be  very  nearly  in 
the  ecliptic,  that  is,  at  or  near  one  of  its  nodes.  Therefore, 
the  conditions  for  a  lunar  eclipse  are  : — 

(1)  The  moon  must  be  in  opposition,  i.e.  full. 

(2)  It  must  be  at,  or  near,  one  of  its  nodes. 

There  are  two  kinds  of  lunar  eclipse,  total  and  partial. 
It  is  total  when  the  whole  surface  of  the  moon  passes 
through  the  shadow,  and  partial  when  only  part  of  its 
surface  is  involved. 

134.  Let  8  and  E  (fig.  67)  represent  the  centres  of  the 
sun  and  earth,  respectively.     Draw  a  pair  of  direct  common 
tangents  AB  and  CD  to  the  sun  and  earth,  meeting  SE 


170  ECLIPSES.  [CHAP.  x. 

produced  in  V,  and  a  transverse  pair  AD  and  BC  meeting 
SE  in  X.  If  these  lines  be  now  supposed  to  revolve  round 
SE  as  axis  they  will  generate  cones,  and  there  is  thus  a 
conical  shadow  BVD,  having  V  as  vertex,  into  which  no 
direct  ray  from  the  sun  can  enter.  This  conical  space  is 
called  the  umbra. 

The  spaces  represented  by  VBL  and  VDN  form  what 
is  called  the  penumbra,  from  which  only  part  of  the  light 
of  the  sun  is  excluded.  It  is  to  be  remembered  that  the 
passage  of  the  moon  through  the  penumbra  does  not  give 
rise  to  any  eclipse,  but  only  to  a  diminution  of  brightness. 


FIG.  67. 

Thus  the  moon  when  at  MI  (fig.  67)  receives  light  from 
portions  of  the  sun  next  A,  but  rays  from  parts  near  C 
will  not  reach  the  moon,  owing  to  the  interposition  of  the 
earth ;  consequently,  the  brightness  of  the  moon  is  some- 
what diminished,  the  diminution  being  greater  the  nearer 
the  moon  approaches  the  edge  of  the  umbra.  An  eclipse, 
properly  so-called,  however,  does  not  commence  until  a  por- 
tion of  the  moon's  surface  shall  have  entered  the  umbra. 

Phenomena  due  to  Refraction. 

135.  As  everyone  who  has  seen  a  total  eclipse  is  aware, 
the  moon  appears  of  a  dull-red  or  brownish  colour.  It 
must,  therefore,  receive  light  from  some  source.  That  it 
is  not  due  to  earth-shine  (Art.  119J  is  certain,  for  the  moon 


CHAP.  X.] 


BREADTH  OF  EARTH'S  SHADOW. 


171 


being  in  opposition,  the  dark  hemisphere  of  the  earth  is 
turned  towards  it.  The  phenomenon  is  caused  by  the 
refracting  power  of  the  earth's  atmosphere,  owing  to  which 
those  rays  from  the  sun,  which  nearly  touch  the  earth, 
are  bent  round,  and  thus  reach  the  moon's  surface. 

Another  curious  phenomenon,  due  to  refraction,  is  seen 
when  an  eclipse  occurs  at  sunset  or  sunrise;  for  both  the 
sun  and  moon  being  elevated  by  refraction,  it  is  possible 
to  see  the  moon  eclipsed  when  the  sun  still  appears  shining 
in  the  heavens,  a  phenomenon  which  was  observed  in  1666, 
1668,  and  1750. 

N. B. —Throughout  the  remainder  of  this  Chapter  we 
shall  occasionally  denote  the  sun  by  the  symbol  O,  and 
the  moon  by  @  . 


To  find  the  Diameter  of  the  Section  of  the  Earth's  Shadow 
where  the  Moon  crosses  it. 

136.  The  angular  diameter  of  the  cross-section  of  the 
cone  of  shadow  is  represented  by  the  arc  MN.  Let  the 
semiangle  MEV  subtended  by  MN  at  the  centre  of  the 
earth  be  a  (fig.  68). 


FIG.  68. 

Let        p  =  0's  hor.  parallax  =  L  EAX  (fig.  68). 
/  =  C  's  hor.  parallax  =  L  EMB  or  L  EXB. 
s  =  angle  subtended  by  O's  semidiameter  at 

E=LSEA. 
6  =  L  E  VB,  the  semiangle  of  cone  of  shadow. 


172  ECLIPSES.  [CHAP.  x. 

Now  by  Euclid  (i.  32)  we  have  :  — 

a+0  =  p';      .-.   a=/-0. 
For  the  same  reason     0  =  s  -  p  ; 

.'.  a  =  p'  -  (s  -  p)  =p'  +  p  -  s. 

But  p,  p',  and  s  are  known  ;  therefore  2a,  the  angle  sub- 
tended by  MN  at  E,  is  determined. 

If  the  moon's  horizontal  parallax  be  taken  as  57',  the 
sun's  as  8",  and  the  sun's  semidiameter  as  16',  the  breadth  of 
the  shadow  2a,  or  2  (p'  +  p  -  s)  is  found  to  be  about  82'.*  As 
the  moon's  angular  diameter  has  a  mean  value  of  about  half 
a  degree,  or  30',  we  see  that  the  breadth  of  the  section  of  the 
shadow  at  the  distance  of  the  moon  is  nearly  three  diameters 
of  the  moon  ;  and  since  the  moon  moves  through  an  arc  equal 
to  its  own  diameter  in  about  an  hour  (Art.  113),  we  see  that 
when  the  moon  passes  through  the  axis  of  the  shadow,  that 
is,  when  the  eclipse  is  central,  it  may  remain  totally  eclipsed 
for  about  two  hours. 

137.  In  the  above  we  see  that  the  semiangle  9  of  the  cone 
=  s  -  p  =  (G's  semidiam.)  -  (O's  hor.  parallax). 

In  the  same  way  as  the  breadth  of  the  section  of  the  cone 
at  MN  has  been  found,  it  can  be  shown  that  the  semidiameter 
of  the  section  at  XY9  where  the  moon  crosses  it  when  in 
conjunction,  is  equal  to  p  -  p  +  s.  For,  by  Euclid  (i.  32), 

p'  +  0=p'  -p  +  s. 


To  find  the  Length  of  the  Earth's  Shadow. 

138.  The  distance  EV  (fig.  68)  from  the  centre  of  the 
earth  to  the  vertex  of  the  cone  is  called  the  length  or  height 
of  the  earth's  shadow.  Its  magnitude  can  now  be  found, 

*  Or  to  be  more  accurate,  the  breadth  of  the  shadow  varies  from  a  maximum 
of  89'  14"  to  a  minimum  value  of  75'  38",  the  maximum  value  being  reached 
when  the  moon  is  nearest  the  earth  (perigee)  and  the  earth  at  the  same  time 
farthest  from  the  sun  (aphelion),  the  minimum  value  being  attained  when 
these  conditions  are  reversed. 


CHAP.  X.]  ECLIPSES    OF    THE    SUN.  173 

knowing  the  earth's  radius  and  the  semiangle  0  of  the  cone. 
For,  since  the  angle  0  is  so  small  (being  equal  to  s  -  p),  we 
may  assume  that  r,  the  radius  of  the  earth,  coincides  with  an 
arc  of  the  circle  whose  centre  is  Pand  radius  VE  (fig.  68). 
Therefore  it  follows  at  once  from  circular  measure  that 

_?"_       JL- 
206265"  ~  EV" 

_  206265"  r  _  206265"  r 

r~      "7'^T7"' 

Taking  r,  the  radius  of  the  earth,  roughly  as  4000  miles, 
s  the  semidiameter  of  the  sun  as  16'  or  960",  and  p,  the  sun's 
parallax  as  8",  we  have 

206265  x  4000 

EV     -"  miles 


=  about  860,000  miles, 
or  215  times  the  earth's  radius. 

Since  the  moon's  distance  from  the  earth  is  only  about 
sixty  times  the  earth's  radius,  we  see  that  the  moon's  orbit 
extends  for  a  much  less  distance  from  the  earth  than  the 
length  of  the  cone  of  shadow,  and  therefore  a  lunar  eclipse 
must  happen  if  the  moon  is  at  one  of  its  nodes  and  full  at  the 
same  time. 

139.  Solar  Eclipses.  —  An  eclipse  of  the  sun  is  caused 
by  the  interposition  of  the  moon  between  the  sun  and  the 
observer.  As  in  the  case  of  a  lunar  eclipse  the  moon  must  be 
nearly  in  the  plane  of  the  ecliptic.  The  two  conditions  for  a 
solar  eclipse  are  therefore  :  — 

(1)  The  moon  must  be  in  conjunction,  i.e.  it  must  be  new 

moon. 

(2)  It  must  be  at,  or  near,  one  of  its  nodes. 

In  a  lunar  eclipse,  the  moon,  on  entering  the  umbra,  loses 
its  light,  and  consequently  the  eclipse  is  visible  from  any  part 
of  the  hemisphere  of  the  earth  which  is  turned  towards  the 
moon.  On  the  other  hand,  in  the  case  of  a  solar  eclipse,  the 


174  ECLIPSES.  ("CHAP.  x. 

light  of  the  sun  is  merely  hidden  from  the  observer ;  and  the 
moon  being  much  smaller  than  the  earth,  this  shows  that  a 
solar  eclipse  can  only  be  visible  over  a  very  limited  area  at 
the  same  time. 

There  may  be  an  eclipse  of  the  sun  visible  from  some 
portion  of  the  earth  if  any  part  of  the  moon  come  within  the 
arc  XY  (fig.  68) ;  and  there  may  be  a  lunar  eclipse  if  it  enter 
MN.  Since  the  arc  XYis  greater  than  JfJVwe  should  expect 
that  more  solar  eclipses  should  occur  than  lunar  if  we  count 
the  eclipses  observed  over  the  whole  earth ;  and  this  in  fact 
is  the  case  ;  but,  as  we  have  just  seen,  an  eclipse  of  the  sun 
is  only  visible  over  a  very  limited  area  of  the  earth,  and 
therefore  it  happens  that  there  are  more  lunar  than  solar 
eclipses  seen  from  any  particular  place. 

140.  There  are  three  kinds  of  solar  eclipses — (1)  total ; 
(2)  annular  ;  and  (3)  partial.  When  the  eclipse  is  total  the 
whole  of  the  sun's  disc  is  hidden  from  view,  whereas  in  the 
case  of  an  annular  eclipse  only  the  central  portion  is  darkened, 
with  a  bright  ring  surrounding  it. 

In  order  to  arrive  at  a  clear  idea  as  to  how  the  moon, 
coming  between  the  sun  and  the  observer,  can  sometimes 
hide  the  whole  of  the  sun's  surface  from  view,  and  at  other 
times  only  the  central  portion,  it  should  be  borne  in  mind 
that  the  moon's  apparent  angular  diameter  is  not  constant, 
for,  on  account  of  its  elliptic  orbit  its  distance  from  the  earth 
is  variable  (Art.  113) ;  the  apparent  diameter  of  the  sun  also 
varies,  the  mean  values  of  both  being  very  nearly  equal. 
The  moon's  angular  diameter  varies  from  33'  22"  when 
nearest  the  earth  (perigee)  to  28'  48"  when  at  its  greatest 
distance  (apogee) ,  and  in  the  case  of  the  sun  the  variation  is 
from  32'  36"  to  31'  32".  When  the  moon's  apparent  diameter 
is  greater  than  that  of  the  sun,  which  occurs  when  it  is 
closest  to  the  earth,  it  will  hide  the  whole  of  the  sun's 
surface  from  the  view  of  an  observer  situated  on  the  line  of 
centres  of  the  two  bodies,  causing  thus  a  total  eclipse.  When, 


CHAP.  X.]  LENGTH  OF  THE  MOON'S  SHADOW.  175 

however,  the  moon's  apparent  diameter  is  less  than  that  of 
the  sun,  as  it  is  when  at  its  greatest  distance  from  the  earth, 
it  will,  under  the  same  conditions,  hide  only  the  central 
portion  of  the  sun,  giving  rise  to  an  annular  eclipse. 

A  very  simple  experiment  will  render  the  ahove  expla- 
nation perfectly  clear.  If  the  reader  take  a  coin,  and,  closing 
one  eye,  hold  it  in  such  a  position  before  the  other  eye  as  to 
just  completely  hide  the  sun's  surface  from  view,  the  position 
of  the  coin  now  is  similar  to  that  occupied  by  the  moon  when 
totally  eclipsing  the  sun.  If,  however,  the  coin  be  removed 
to  a  greater  distance  from  the  eye,  keeping  its  centre  still  in 
a  direct  line  with  that  of  the  sun,  it  will  be  found,  owing  to 
the  diminution  in  its  apparent  diameter,  due  to  the  increase 
of  distance,  that  it  only  hides  the  central  portion  of  the  sun 
from  view,  thus  illustrating  how  the  moon,  when  farthest 
from  the  earth,  causes  an  annular  eclipse. 

When  a  partial  eclipse  takes  place,  only  a  portion  of  the 
sun's  disc  at  one  side  becomes  darkened,  owing  to  the  centres 
of  the  two  bodies  not  being  in  a  direct  line  with  the  observer. 
It  is  evident  that  all  total  and  annular  eclipses  must  begin 
and  end  as  partial  eclipses. 

To  find  the  Length  of  the  Cone  of  Shadow  cast  by  the  Moon. 

141.  Let  S  denote  the  centre  of  the  sun,  M  of  the  moon, 
and  R  and  r  the  radii  of  the  sun  and  moon  respectively 
(fig.  69).  The  apex  of  the  shadow  cast  by  the  moon  will 
be  at  0  where  the  common  tangents  CH  and  DFmeet.  It 
is  required  to  find  the  distance  MO. 

Since  the  triangles  OSC  and  OMH  are  similar  we  have, 

Euclid  (vi.  4), 

OS      JR     .,  OM+SM     jR. 


therefore  solving  for  OM  and  denoting  SM  by  d  we  have 


176  EGXIPSES.  [CHAP.  x. 

But  r,  the  radius  of  the  moon,  is  about  1076  miles,  while 
d,  the  distance  between  the  centres  of  the  sun  and  moon 
varies  from  11,717  to  11,713  times  the  earth's  diameter. 
Substituting  these  values  we  find  that  OH  varies  from  28-94 
to  28 -93  times  the  earth's  diameter. 


FIG.  69. 

Since  the  distance  from  the  moon's  centre  to  the  surface 
of  the  earth  varies  from  28  to  31  diameters  of  the  earth,  it 
follows  that  the  observer  may  sometimes  be  situated  at  E 
(fig.  69)  inside  the  cone  of  shadow,  and  at  other  times  at  E\ 
beyond  the  point  0,  where  the  cone  tapers  to  a  vertex.  In 
the  former  case  a  total  eclipse  takes  place,  the  moon  subtend- 
ing a  greater  angle  than  the  sun,  and  in  the  latter  case  an 
annular  eclipse  occurs,  the  portion  of  the  sun's  surface  hidden 
from  view  being  represented  by  the  inner  circle  QN  (fig.  69j, 
marked  off  by  tangents  drawn  from  E  to  the  surface  of  the 
moon,  and  produced  out  to  meet  the  sun. 

To  calculate  the  Conditions  for  a  Lunar  or  Solar  Eclipse. 

142.  Lunar  Eclipse. — Let  0  (fig.  70)  represent  the 
centre  of  the  section  of  the  earth's  shadow  at  the  distance 
of  the  moon,  and  M  the  centre  of  the  moon  when  touching 
the  shadow  externally.  MN  represents  the  apparent  path 
of  the  moon,  2^0  the  ecliptic,  and  N  the  position  of  the 
node. 


CHAP.  X.]  CONDITIONS  FOR  AN  ECLIPSE. 


177 


Now  it  is  evident  that  an  eclipse  of  no  portion  of  the 
moon's  surface  can  take  place  unless  the  distance  between 
the  centres  of  the  moon  and  shadow  becomes  less  than  MO. 


FIG.  70. 


But  MO  =  (semidiam.  of  shadow)  +  (semidiam.  of  moon) 


m. 


But 


where 


a=p'+p-s  (Art.  136); 
MO  =  p'  +  p-s  +  m, 

p  =  O's  hor.  parallax  =  8", 
/  =  's  hor.  parallax  =  57', 
s  =  O's  semidiam.  =  16'  (mean  value), 
m  =  C's  semidiam.  =  15'  (mean  value). 
Therefore,  we  have 

MO  =  57'  +  8"  -  16'  +  15'  =  56'  (roughly). 
Similarly,  for  a  total  lunar  eclipse  the  moon  will,  in  the 
limiting  position,  touch  the  shadow  internally,  and  we  shall 
have  — 

MO  =  (radius  of  shadow)  -  (semidiam.  of  moon) 

=  a-  m 

*=p'+p-s-m  =  26'  (roughly). 

Therefore  it  is  impossible  for  a  lunar  eclipse  to  occur  if 
the  distance  between  the  centres  of  the  moon  and  shadow 
exceeds  56',  and  for  a  total  eclipse  the  distance  cannot 
exceed  26'. 

143.  Solar  Eclipse.-—  We  have  seen  (Art.  137)  that  the 
angular  radius  of  the  section  of  the  cone  where  the  moon 

N 


178  ECLIPSES.  LCHAP.  x. 

crosses  it  in  conjunction  at  XF(fig.  68)  isp'-p+  s;  there- 
fore, it  is  evident  that  the  limiting  distance  of  the  moon 
from  the  centre  of  the  section  for  a  partial  eclipse  of  the 
sun  is  pr  -  p  +  s  +  m  or  about  88',  the  limiting  distance  for 
a  total  eclipse  being  p'  -p  +s  -  m  or  58'. 

As  the  moon's  orbit  is  inclined  at  such  a  small  angle 
to  the  ecliptic  (5°),  the  distance  MO  must  be  nearly  per- 
pendicular to  the  ecliptic,  and  therefore  is  almost  equal  to 
the  latitude  of  the  moon ;  but,  as  the  latitude  of  the  moon 
varies  from  0°  to  5°,  we  see  that  an  eclipse  can  only  take 
place  very  near  a  node. 

144.  Definition. — The  greatest  distance  (measured  along 
the  ecliptic)  of  the  moon  from  the  node,  when  in  opposition, 
at  which  an  eclipse  can  happen  is  called  the  Lunar  Ecliptic 
Limit.     Thus,  in  fig.  71,  the  apparent  path  of  the  moon 
is  represented  by  MN,  the  moon  being  taken  just  touching 
the  shadow  when  nearly  in  opposition ;  then  the  distance 
NO  will  represent  the  distance,  measured  along  the  ecliptic, 
of  the  moon  from  the  node  when  in  opposition  (i.e.  the  pro- 
jection on  the  ecliptic  of  the  moon's  distance  from  the  node 
when  in  opposition),  and  is  therefore  the  ecliptic  limit. 

To  find  the  Lunar  Ecliptic  Limit. 

145.  In  order  to  calculate  the  ecliptic  limit  NO,  in  the 


N 


FIG.  71. 


spherical  triangle  MON,  the  arc  MO  is  known,  being  the 
sum  of  the  semidiameters  of  the  shadow  and  the  moon  (by 


CHAP.  X.]  ECLIPTIC  LIMITS.  179 

Art.  142,  M 0  =  p'  +  p  -  s  +  m) ;  the  angle  N,  the  inclination 
of  the  moon's  orbit  to  the  ecliptic  is  also  known,  being  about 
5°,  and  the  angle  M  is  a  right  angle  (since  OM is  the  shortest 
distance  from  0  to  MN) ;  therefore,  the  arc  NO  can  be 
calculated. 

Major  and  Minor  Limits.— The  lunar  ecliptic  limit 
is  not  a  constant  quantity,  as  the  parallax  and  semidiameter 
of  both  the  sun  and  moon  are  variable.  Moreover,  the 
inclination  of  the  moon's  orbit  varies  from  5°  20'  to  4°  57'. 
All  these  causes  combine  to  produce  considerable  variations 
in  the  limit.  When  the  moon  is  nearest  the  earth  and  the 
earth  farthest  from  the  sun,  and  at  the  same  time  the  angle 
of  inclination  of  the  moon's  orbit  least,  the  circumstances 
are  then  most  favourable  for  an  eclipse,  which  may,  therefore, 
take  place  at  a  greater  distance  from  the  node  than  at  any 
other  time.  Under  these  circumstances  the  magnitude  of 
ON  is  found  to  be  12°  5',  and  is  called  the  Major  Ecliptic 
Limit. 

On  the  other  hand,  when  the  moon  is  farthest  from  the 
earth,  the  earth  nearest  the  sun,  and  the  angle  at  N  (fig.  71) 
greatest,  the  circumstances  are  most  unfavourable  for  an 
eclipse,  and  the  moon  must  be  much  closer  to  the  node  than 
in  the  former  case,  in  order  that  an  eclipse  should  occur. 
Under  these  conditions  ON  is  found  to  be  9°  30',  and  is 
called  the  Minor  Ecliptic  Limit. 

When  the  distance  of  the  moon  from  the  node  at  oppo- 
sition is  within  the  major  limit  the  eclipse  may  take 
place,  but  within  the  minor  limit  it  must  take  place. 

146.  Solar  Ecliptic  Limits. — There  is  also- an  ecliptic 
limit  for  the  sun,  viz.  the  greatest  distance  (measured  along 
the  ecliptic)  of  the  moon  from  the  node,  when  in  conjunction, 
consistent  with  a  solar  eclipse.  The  maximum  and  mini- 
mum values  are  also  called  the  major  and  minor  limits  ;  the 


180  ECLIPSES.  [CHAP.  x. 

former  being  18°  31'  within  which  a  solar  eclipse  may  take 
place,  and  the  latter  15°  2 1/  within  which  it  must  occur. 

147.  In  (Art.  126)  it  was  seen  that  the  moon's  nodes 
have  a  retrograde  motion  on  the  ecliptic,  making  a  complete 
revolution  in  18-f  years.  From  this  it  was  proved  (Art.  127), 
that  the  synodic  period  of  revolution  of  the  line  of  nodes 
is  346*62  days,  or,  in  other  words,  we  might  say  that  the 
sun  separates  from  the  line  of  nodes  through  360°  in  346'62 
days.  Therefore,  in  one  synodic  lunar  month  of  29 J  days, 
the  sun  separates  from  a  node  through  an  angle 

30°  38'  =30°  I  nearly. 


346-62 

As  the  comparison  of  this  result  with  the  solar  and  lunar 
ecliptic  limits  enables  us  to  calculate  the  frequency  of 
eclipses,  it  is  of  importance  that  the  student  should 
remember  the  approximate  values  of  the  following  quan- 
tities : — 

Major.        Minor. 

Lunar  ecliptic  limits,       , , »  ;    12°          9°  J. 
Solar  ecliptic  limits,          .      18°|       15°|. 
Eelative  motion  of  sun       =    30°  |  in  each  lunation. 
Period  of  sun's  revolution  =  346  days. 
Period  from  node  to  node  =  173  days. 
Six  lunations  =  6  x  29  i       =177  days. 

N.B. — It  is  evident  that  during  either  a  lunar  or  solar 
eclipse  the  distances  of  the  sun  and  moon  from  the  nearest 
node  are  nearly  equal. 

To  determine  the  Frequency  of  Eclipses. 

148.  Least  Possible  dumber.— Let  N  and  n  repre- 
sent the  moon's  nodes  (fig.  72).  Cut  off,  on  the  ecliptic  EC, 
distances  NL,  NL',  nl,  nl'  each  equal  to  the  lunar  ecliptic 


€HAP.  X.]  FREQUENCY  OF  ECLIPSES.  181 

limit ;  and  similarly  NS,  NS',  ns,  m'  each  equal  to  the  solar 
ecliptic  limit.  Now  as  the  sun  moves  with  reference  to  the 
nodes  through  30°f  in  one  synodic  month,  it  follows  that 
he  will  take  more  than  a  month  in  moving  through  the  arc 
SS'  or  ss';  for  the  least  value  of  these  arcs,  being  double  the 
sun's  minor  limit,  is  31°.  Hence  at  least  one  new  moon, 
and  therefore  one  solar  eclipse,  must  occur  within  each  of 
these  arcs. 


FIG.  72. 

On  the  other  hand,  the  least  value  of  LL'  or  U,  being 
twice  the  moon's  minor  limit,  is  only  19°,  and  the  sun 
traverses  each  of  these  arcs  in  much  less  than  a  month 
(about  18  days)  ;  therefore,  it  is  possible  that  there  may 
be  no  full  moon  near  either  node,  and  therefore  no  lunar 
eclipse  during  the  year.  Hence  the  least  possible  number  of 
eclipses  in  a  year  is  two,  both  of  the  sun. 

149.  Greatest  Possible  Number. — The  sun  takes 
173  days  to  pass  from  N  to  n  (fig.  72),  or  4  days  less  than 
six  lunations  (177  days) ;  therefore,  when  the  moon  happens 
to  be  full  two  days  before  the  sun  arrives  at  N,  there  will 
also  be  a  full  moon  2  days  after  his  passage  through  n,  thus 
rendering  a  lunar  eclipse  very  close  to  each  node  a  certainty. 


182  ECLIPSES.  [CHAP.  x. 

But  if  a  lunar  eclipse  occur  2  days  before  or  after  the  sun's 
passage  through  a  node,  there  may  also  be  two  solar  eclipses 
near  that  node,  viz.  at  the  preceding  and  following  new 
moon ;  for,  in  half  a  lunation,  or  14f  days,  the  sun  moves 
through  15°-^;  and  even  if  to  this  we  add  the  arc  gone 
through  in  2  days,  the  result  is  still  well  within  the  sun's 
major  ecliptic  limit.  Therefore,  there  may  be  one  lunar 
and  two  solar  eclipses  at  each  node  in  a  period  of  346  days. 
But  if  the  eclipses  within  SS'  (fig.  72)  occur  in  January, 
there  will  be  ample  time  before  the  year  is  completed  for 
the  sun  to  arrive  a  second  time  within  SS'.  There  will 
now  be  another  solar  eclipse  near  S,  followed  by  a  lunar 
eclipse  6  days  after  the  sun's  passage  through  JV".  There  will 
not  now,  however,  be  a  solar  eclipse  near  /S',  as  the  following 
new  moon  will  take  place  outside  the  sun's  major  limit.  In 
all,  we  have  counted  eight  eclipses  in  12J  lunations  or  368 
days,  viz.  five  of  the  sun  and  three  of  the  moon.  But  all 
these  eight  eclipses  cannot  happen  in  a  year  (365  days)  ; 
therefore  one,  either  a  solar  or  lunar,  will  have  to  be  omitted. 
Hence  the  greatest  possible  number  of  eclipses  in  a  year  is  seven, 
five  of  the  sun  and  two  of  the  moon,  or  four  of  the  sun  and  three 
of  the  moon. 

In  every  18  years  there  are  generally  41  eclipses  of  the 
sun  to  29  of  the  moon. 

Chaldean  Saros. 

150.  The  synodic  period  of  the  moon's  nodes  being 
346-62  days,  and  a  lunation  being  29*53  days,  we  therefore 
have — 

19  synodic  revolutions  of  node  =  19  x  346-62  days 

=  6585  days ; 
also        223  lunations  =  223  x  29-53  =  6585  days. 

Therefore,  we  see  that  after  every  period  of  6585  days, 
which  are  equivalent  to  18  years  11  days  or  18  years 


CHAP.  X.]  SAROS  OF  JTHE  CHALDEANS.  183 

10  days,  according  as  there  are  four  or  five  leap  years  in 
the  interval,  the  sun  and  moon  will  return  to  nearly  the 
same  positions  relative  to  the  nodes  (each  having  made  an 
exact  numher  of  revolutions),  and  therefore  the  eclipses 
will  repeat  themselves  in  the  following  cycle  in  the  same 
order  as  in  the  previous  one.  This  period  is  called  the 
Chaldean  Saros,  as,  by  means  of  it,  the  Chaldeans  wero 
enabled  to  foretell  the  occurrence  of  eclipses. 


(    184    ) 


CHAPTER  XI. 

TIME. 

Mean  and  Apparent  Time.    Equation  of  Time. 

151.  We  explained  in  Chapter  III.  the  difference  be- 
tween sidereal  and  solar    time.      The  sidereal   day  is  of 
constant  length  as  the  rotation  of  the  earth  on  its  axis  is 
uniform.     The  length  of  the  apparent  solar  day,  however,  is 
variable,  as  the  sun's  rate  of  change  of  right  ascension  is  not 
uniform  throughout  the  year.     On  account  of  this  inequality 
a  clock  cannot  be  regulated  to  point  to  12  o'clock  j  ust  when 
the  sun  is  in  the  meridian.      Accordingly  our  clocks,  instead 
of  keeping  apparent  solar  time  keep  mean  solar  time  as  indi- 
cated by  the  motion  of  an  imaginary  body  called  the  mean 
sun,  which  is  supposed  to  move  uniformly  in  the  equator  at 
the  same  mean  rate  as  that  of  the  true  sun  in  the  ecliptic. 

Definition. — A  mean  solar  day  is  the  interval  between 
two  successive  transits  of  the  mean  sun  across  the  meri- 
dian. 

As  the  mean  sun  changes  its  right  ascension  at  a  uniform 
rate  we  see  that  the  length  of  a  mean  solar  day  is  constant. 
The  hour-angle  of  the  mean  sun  at  any  instant  measures  the 
mean  time  at  that  instant,  while  that  of  the  apparent  or  real 
sun  gives  the  apparent  time,  or  time  as  indicated  by  a  sun- 
dial. 

152.  Definition. — The  equation  of  time  is  the  difference 
between  the  mean  and  the  apparent  time.     It  is^coAinted 
positive  when^the  mean  exceeds  the   apparent  time,  and 


CHAP.  XI.]  EQUATION    OF   TIME.  185 

negative  when  the  latter  exceeds  the  former.     Therefore  we 
have — 

(Mean  Time)  -  (Apparent  Time)  =  (Equation  of  Time), 
or     (Clock  Time)  -  (Dial  Time)  =  (Equation  of  Time). 

As  the  true  sun  moves  in  the  ecliptic  and  the  mean  sun 
in  the  equator,  the  motion  of  the  former  being  variable,  and 
of  the  latter,  uniform,  we  see  that  the  equation  of  time  is  due 
to  two  causes : — 

(1)  The  variable  motion  of  the  true  sun  in  the  ecliptic  owing 

to  the  eccentricity  of  the  earth's  orbit. 

(2)  The  obliquity  of  the  ecliptic  to  the  equator. 

We  shall  now  consider  each  of  these  causes  separately,  and 
by  combining  their  effects  we  shall  be  able  to  see  how  the 
equation  of  time  varies  throughout  the  year,  when  it  reaches 
a  maximum,  and  at  what  periods  it  vanishes. 

Equation  of  Time  due  to  Unequal  Motion  of  Sun. 

'  153.  On  December  31st,  at  perihelion,  or,  in  other  words, 
when  the  earth  is  nearest  the  sun,  its  velocity  is  greatest 
(Art.  68),  and  therefore  the  rate  at  which  the  sun  moves  from 
west  to  east  along  the  ecliptic  will,  at  this  period,  be  greater 
than  the  mean  rate.  As  the  earth  turns  on  its  axis  from 
west  to  east  this  would  cause  the  apparent  or  true  solar  days 
to  exceed  in  length  the  mean  solar  days,  and,  if  a  sun-dial 
and  a  clock  be  started  together  at  perihelion,  the  apparent  time 
will  gradually  get  behind  mean  time,  so  that  the  sun-dial  will 
lose  compared  with  the  clock.  This  will  continue  for  about 
three  months  until  the  rate  of  the  sun  in  the  ecliptic  becomes 
equal  to  its  mean  rate.  Hence  that  component  of  the  equa- 
tion of  time,  due  to  the  unequal  motion  of  the  sun,  reaches 
at  the  end  of  March  its  greatest  positive  value,  viz.  about 
7  minutes.  The  sun-dial  will  now  begin  to  gain  on  the 
clock  what  it  lost  in  the  preceding  three  months,  until 
aphelion  (July  1st)  is  reached,  when  it  will  coincide  with 


186  TIME.  [CHAP,  xi 

the  clock,  and  the  equation  of  time,  in  so  far  as  it  is  due  to 
the  unequal  motion  of  the  sun,  vanishes. 

Similarly  we  can  see  that  from  aphelion  to  perihelion  the 
equation  of  time,  due  to  this  cause,  is  negative,  having  a 
maximum  value  of  -  7  minutes  at  the  end  of  September. 

Equation  of  Time  due  to  the  Obliquity  of  the  Ecliptic. 

154.  Even  if  the  sun's  motion  along  the  ecliptic  were 
uniform,  his  rate  of  change  in  right  ascension  would  still  be 
variable  on  account  of  the  obliquity  of  the  ecliptic  to  the 
equator.  Let  us  now  suppose  that  the  true  sun  8  and  the 
mean  sun  /Si  (fig.  73)  start  together  at  the  vernal  equinox  r, 
the  former  moving  in  the  ecliptic  and  the  latter  in  the 
equator;  they  will  again  be  together  at  the  autumnal 
equinox  =c=,  and  also  their  right  ascensions  will  coincide  at  the 
two  solstices.  Hence  that  portion  of  the  equation  of  time 
due  solely  to  the  obliquity  of  the  ecliptic  becomes  zero  four 
times  each  year,  at  the  equinoxes  and  solstices. 


FIG.  73. 


Again,  when   the   true  sun  is  at  S  (fig.   73)  his  right 
ascension  is  r  K  (PSK  being  the  arc  passing  through  the 


CHAP.  XI.]  EQUATION  OF  TIME.  187 

celestial  pole  and  the  sun's  centre.  But  the  position  of  the 
mean  sun  Si  (as  affected  by  the  obliquity  of  the  ecliptic 
alone)  would  be  obtained  by  cutting  off  T  Si  =  T  $,  Si  falling 
to  the  east  of  K,  since  T  S,  being  the  hypotenuse  of  the 
right-angled  spherical  triangle  T  SK,  is  greater  than  T  K. 
Therefore  the  true  sun,  being  to  the  west  of  the  mean  sun, 
will  each  day  cross  the  meridian  first,  the  sun-dial  being  faster 
than  the  clock.  Hence  this  portion  of  the  equation  of  time 
is  negative,  its  greatest  value  being  about  -  10  minutes. 
Similarly  we  may  see  that  from  solstice  to  equinox  the  dial 
will  be  slower  than  the  clock,  and  the  component  of  the  equa- 
tion of  time  will  be  positive,  having  a  maximum  value  of  + 10 
minutes. 


Combination  of  the  Two  Components. 

155.  Let  X  =  that  portion  of  equation  of  time  due  to  the 
unequal  motion  of  the  sun. 

T  =  that  due  to  obliquity  of  the  ecliptic. 
If  we  summarize  the  above  results  we  have — 

(1)  X  vanishes  twice  each  year,  on  December  31st  and 
July  1st,  and  varies  from  a  maximum  of  +  7  minutes  at  end 
of  March  to  -  7  minutes  at  end  of  September.     (See  dotted 
curve,  fig.  74.) 

(2)  Y  vanishes  four  times  each  year,  at  the  equinoxes  and 
solstices.     From  equinox  to  solstice  T  is  negative,  and  from 
solstice  to  equinox,  positive,  varying  from  a  maximum  of 
+  10  minutes  to  -  10  minutes  at  intermediate  points.     (See 
continuous  curve,  fig.  74.) 

(3)  The  sum  or  difference  of  X  and  Y,  according  as  they 
are  of  the  same  or  opposite  sign,  gives  the  equation  of  time 
at  any  instant.     (See  curve,  fig.  75.) 


"FnrH*         \ 

{UNIVERSITY! 


188  TIME.  [CHAP.  xi. 

The  Equation  of  Time  vanishes  four  times  each  Year. 

156.  We  have  seen  that  the  equation  of  time  is  equal  to 
the  algebraic  sum  of  X  and  F;  the  maxima  values  of  T being 

+  lOw,     -  10m,     +  10m,     -  10w, 
which  occur  in  the  months 

February,  May,  August,  November. 
Now  as  X  has  never  a  greater  numerical  value  than  ±  7 
minutes  it  follows  that,  in  the  four  months  mentioned  above, 
the  equation  of  time  (X  +  Y)  must  have  the  same  sign  as  Y, 
whether  X  is  positive  or  negative.  Hence  it  follows  that  the 
equation  of  time  has  at  least  four  changes  of  sign  throughout 
the  year,  viz. :  +>  _.  +>  _? 

and  therefore,  on  changing  from  positive  to  negative  or  vice 
versa,  must  at  least  on  four  occasions  pass  through  a  zero 
value.* 

The  dates  on  which  the  equation  of  time  vanishes  are 
about  April  16,  June  15,  September  1,  and  December  25. 
The  greatest  positive  value  is  14m  28s  on  February  11,  and 
the  greatest  negative  value  16m  21s  on  November  3.  (See 
the  curve  in  fig.  75.) 

157.  We  can  now  represent  graphically  how  the  equation 
of  time  varies  throughout  the  year,  when  it  reaches  a  maxi- 
mum, and  at  what  periods  it  vanishes. 


FIG.  74. 

These  curves  represent  the  variations  in  the  two  components  of  the 
equation  of  time. 

*  That  it  does  not  vanish  oftener  than  four  times  each  year  appears  from, 
an  examination  of  the  curve  in  fig.  75. 


CHAP.  XI.]       MORNING  AND  AFTERNOON  UNEQUAL.  189 

In  fig.  74  the  dotted  curve  represents  the  component  due 
to  the  unequal  motion  of  the  sun,  and  the  continuous  line 
that  due  to  the  obliquity  of  the  ecliptic.  In  fig.  75  is  a  single 
curve  representing  the  combined  effect  of  the  other  two,  the 
equation  of  time  corresponding  to  any  point  on  the  curve 
being  represented  by  the  perpendicular  distance  of  the  point 
from  the  zero  line.  Thus  the  equation  of  time  corresponding 
to  p  is  represented  by  pm.  All  portions  of  the  curve  below 
the  zero  line  represent  negative  values.  The  periods  at  which 
the  equation  of  time  vanishes  are  represented  by  the  points 
where  the  curve  cuts  the  zero  line. 


FIG.  75. 


The  curve  represents  the  variations  in  the  equation  of  time  found  by  combining 
the  two  components  whose  curves  are  given  in  the  preceding  figure. 


Morning  and  Afternoon  unequal  in  Length. 

158.  The  interval  of  time  from  sunrise  until  the  sun  is 
in  the  meridian  (apparent  noon)  is  equal  to  the  interval  from 
apparent  noon  to  sunset,  neglecting  the  small  change  in 
the  sun's  declination  throughout  the  day.  But  mean  and 
apparent  noon  do  not  in  general  coincide  ;  therefore,  morning 
and  afternoon,  as  measured  by  our  clocks,  are  not  of  equal 
length,  the  former  being  less  (algebraically)  than  half  the 
interval  between  sunrise  and  sunset  by  the  equation  of  time, 
the  latter  being  greater  (algebraically)  by  the  same  amount. 


190  TIME.  [CHAP.  xi. 

Hence  the  lengths  of  the  morning  and  afternoon  always 

differ  by  twice  the  equation  of  time,  so  that 

(length  afternoon)  -  (length  morning)  =  2  (equation  of  time). 

Immediately  after  the  winter  solstice,  the  afternoons  begin 
to  lengthen,  while  the  mornings  still  continue  to  get  shorter. 
The  explanation  of  this  is  simple. 

For  the  sun  being  at  the  winter  solstice  he,  as  it  were, 
stands  still  for  some  days,  during  which  time  we  may  regard 
his  declination  as  constant.  The  interval  between  apparent 
noon  and  sunset,  therefore,  remains  constant.  But  as  the 
equation  of  time  is  at  this  period  increasing  [see  curve, 
fig-  75),  mean  noon  precedes  apparent  noon  by  a  greater 
amount  each  day.  Hence  the  mean  time  of  sunset  increases 
and  the  afternoons  get  longer. 

Similarly,  it  may  be  shown  that  while  the  apparent 
time  of  sunrise  remains  the  same,  the  mean  time  of  sun- 
rise increases  each  day,  and,  consequently,  the  mornings 
are  shortened.  It  is,  however,  to  be  borne  in  mind  that 
very  soon  the  increasing  declination  of  the  sun  causes  the 
mornings  to  lengthen  as  well  as  the  afternoons. 

EXAMPLES. 

1.  Given  mean  time  =  5h  12m  20s  p.m.,  and  the  equation  of  time  =-f  5m  25s ; 
find  apparent  time. 

Am.  5h  6m  55s  p.m. 

2.  Given  apparent  time  =  10h  4m  15s  a.m.,  on  November  3,  when  the 
equation  of  time  has  its  greatest  negative  value,  viz.  16m  21s ;  find  mean  time. 

Am.  9h  47m  54s  a.m. 

3.  Find  th'e  meantime  of  apparent  noon  in  questions  1  and  2. 

Am.     1.     5m  25s  p.m. 

2.     Ilh43m  39s  a.m. 

4.  On  Nov.  3,  the  sun-dial  is  16m  21s  faster  than  the  clock.     Given  that 
the  sun  rose  at  6h  57m  a.m. ;  find  the  time  of  sunset. 

Am.  4h  30m  18s  p.m. 

5.  Given  that  the   sun  rose   on  a  certain  date  6h  54m  a.m.,  and  set  at 
4h  33m  p.m.  ;  find  the  equation  of  time. 

Am.  -  16m  30s. 

6.  In  question  5,  by  how  much  does  the  length  of  the  morning  exceed  the 
length  of  the  afternoon  ? 

Am.  33m. 


CHAP.  XI.]  IFFERENCE  OF  LOCAL  TIMES.  191 

Local  Time. 

159.  As  the  earth  rotates  uniformly  on  its  axis  from 
west  to  east,  it  is  evident  that  the  further  east  a  place  is 
situated  the  sooner  will  the  sun  cross  the  meridian  of  that 
place,  and,  therefore,  the  later  will  be  the  local  time. 

When  it  is  noon  at  any  place  it  will  be  1  o'clock  p.m. 
15°  to  the  eastward,  and  11  o'clock  a.m.  15°  to  the  westward 
of  that  place.  For : — 

360°  correspond  to  24  hours  ; 
15°         „          to     1  hour. 

Given  the  longitudes  of  two  places  (A  and  B),  and  the 
time  at  one  of  them  (A),  to  find  the  time  at  the  other  (£). 

Rule. — Divide  the  algebraic  difference  of  the  longitudes  by 
15,  which  gives  the  difference  of  the  local  times.  Add  this  remit 
to,  or  subtract  from ,  the  given  time  at  A,  according  as  B  is  east 
or  west  of  A,  and  the  result  will  be  the  time  at  B. 

EXAMPLE.— Find  the  time  at  New  York  (long.  74°  10'  W.)  when 
it  is  3  o'clock,  P.M.  at  Dublin  (long.  6°  20'  W.). 

Here  the  difference  of  longitudes  =  67°  50'. 
On  dividing  by  15,  we  get : — 

Difference  in  times  =  4h  31m  20". 

New  York  being  west  of  Dublin,  the  time  is  earlier.  There- 
fore, subtract  4h  31m  20s  from  3  p.m.,  that  is,  from  15  hours. 

H.  M.  S. 

15         0         0 
4       31       20 


New  York  time  =  10       28       40  a.m. 

Should  one  longitude  be  east  and  the  other  west,  their 
algebraic  difference  will  be  got  by  adding  the  longitudes. 


192  TIME.  [CHAP.  xi. 

-ZV..Z?. — In  the  same  way,  when  given  the  sidereal  time  at 
one  of  two  places  whose  longitudes  are  known,  the  sidereal 
time  at  the  other  place  can  be  found ;  for  the  earth  turns  on 
its  axis  through  360°  relative  to  the  fixed  stars  in  24  sidereal 
hours,  and  therefore  15°  relative  to  the  fixed  stars  correspond 
to  1  sidereal  hour. 

To  reduce  a  given  Interval  of  Mean  Time  to  Sidereal  Time, 
and  vice  versa. 

160.  There  are  365^  mean  solar  days  in  the  year,  and 
366|-  sidereal  days,  the  sun  making  one  less  diurnal  revolu- 
tion than  the  fixed  stars,  on  account  of  his  annual  motion  in 
the  ecliptic ; 

.*.     365|  mean  solar  days  =  366|  sidereal  days. 
Therefore,  if  m  be  any  interval  of  mean  time,  and  s  the 
corresponding  interval  of  sidereal  time,  we  have— 

365|:  366J-:  :  m:s.v 
From  which,  if  m  be  given,  s  can  be  found,  and  vice  versa. 

EXAMPLES. 

1.  Express  in  sidereal  time  an  interval  of  16h  15m  23s  mean  time. 

Am.  16h  18m  3s. 

2.  Express  in  mean  time  an  interval  of  12h  16m  26s  sidereal  time. 

Ans.  12h  14ra  16". 

N*B. —  We  cannot  convert,  by  means  of  the  above 
formula,  the  actual  mean  time  at  any  instant  in  sidfereal 
time,  or  vice  versa.  This  is  done  as  follows : — 


Given  the  Sidereal  Time  at  any  instant  at  Greenwich,  to  find 
the  Mean  Time  at  that  instant. 

161.  Let  AQ  (fig.  76)  represent  the  meridian,  r<2  the 
celestial  equator,  and  m  the  mean  sun. 


CHAP.  XI.] 

Then 


193 


MEAN  AND  SIDEREAL  TIME. 

Qr  =  sidereal  time  (expressed  in  arc), 
Qm  =  mean  time  (expressed  in  arc), 
m<r  =  right  ascension  mean  sun. 
Qm  =  QT  -  m<r. 

Therefore   the   following   equation  holds  good,  all  the 
quantities  being  supposed  reduced  to  the  same  units : — 

(Mean  time)  =  (Sidereal  time)  -  (E.  A.  mean  sun). 

A 


But 


FIG.  76. 

The  "  Nautical  Almanac  "  contains  the  right  ascension  of  the 
mean  sun  at  noon  for  each  day  at  Greenwich.  The  right 
ascension  of  the  mean  sun  at  noon  is  evidently  the  same  as 
the  sidereal  time  of  mean  noon. 

Therefore,  to  reduce  sidereal  to  mean  time  at  any  instant 
at  Greenwich,  we  have  the  following  rule  : — 

From  the  given  sidereal  time  subtract  the  right  ascension  of 
the  mean  sun  at  noon,  and  reduce  the  result  to  mean  time. 

Similarly,  when  we  are  given  the  mean  time  at  any 
instant  to  find  the  sidereal  time  at  that  instant,  we  have  the 
rule :  — 

Reduce  the  mean  time  to  sidereal  hours,  minutes,  fyc.,  add 
this  to  the  right  ascension  of  the  mean  sun  at  noon,  and  the 
result  is  the  required  sidereal  time. 

o 


194  TIME.  [CHAP,   xi 

The  right  ascension  of  the  mean  sun  at  mean  noon  has 
to  be  corrected  by  allowing  for  the  change  in  the  position  of 
the  mean  sun  in  the  interval  which  has  elapsed  since  mean 
noon.  This  change  is  at  the  rate  of  24  sidereal  hours  in 
365^  days,  or  3m  56S<55  daily  or  9s-8565  in  1  mean  solar 
hour. 


EXAMPLES. 

(1)  Given  mean  time  =  3h  20  n  50s  p.m.,  when  the  right  ascension 
of  the  mean  sun  at  mean  noon  is  16h  32m  9s ;  find  sidereal  time. 

Sidereal  time  =  Mean  time  (riduced)  +  R.  A.  mean  sun. 

H.     M.     s. 

Here  R.  A.  mean  sun  at  noon  =    16     §2     9 
Change  in  3h  20m  50s  =      0       0  34 


16     32  43 
Mean  time  (reduced  Art.  160)  =      3     21  23 


.'.     Sidereal  time  =     19     54     6 

(2)  Given  sidereal  time  =  5h  32m  37%  and  the  right  ascension  of 
the  mean  sun  at  mean  noon  =  7h  37m  32s ;  find  mean  time. 

Here,  mean  time  =  sidereal  time  -  R.  A.  mean  sun. 

H.     M.      s.        H. 

Sidereal  time  =      5     32     37  (+  24) 

B.  A.  mean  sun  at  noon  =      7     37     32 


.*.     Mean  time  =    21     55       5 

(expressed  in  sidereal  hours,  &c.) 

Reducing  this  result  to  mean  hours,  minutes,  &c.  (Art.  160),  we 
get  the  mean  time  =  21h  51m  30s.  But  this  has  to  be  corrected  by 
allowing  for  the  change  in  the  right  ascension  of  the  mean  sun  since 
the  previous  noon,  which  would  amount  to  3m  35s  (approximately). 
Subtracting  this  from  the  previous  result  we  get  the  mean  time 
=  21h  47ffi  55s  (approximately). 


CHAP.  XI.]  MEAN  AND  SIDEREAL  TIME.  195 

To  convert  the  sidereal  time  at  a  given  meridian,  not  that  of 
Greenwich,  into  the  corresponding  mean  solar  time,  and  vice 
versa. 


162.  As  the  "Nautical  Almanac"  only  gives  the  right 
ascension  of  the  mean  sun  for  mean  noon  at  Greenwich, 
therefore,  for  any  other  meridian,  we  will  have  to  proceed 
as  follows. 

Rule. — Reduce  the  given  time,  whether  it  be  sidereal  or 
mean  solar,  to  the  corresponding  Greenwich  time  (Art.  159). 
Then,  knowing  the  It.  A.  of  mean  sun  at  noon,  the  Greenwich 
sidereal  time  can  be  changed  as  above  into  mean  solar  time,  or 
vice  versa,  and  the  result  reduced  back  again  to  the  meridian  of 
the  given  place. 

EXAMPLE.  — Assuming  that  on  a  certain  day  at  Greenwich  the 
right  ascension  of  the  mean  sun  was  10  hours  at  12  o'clock,  find 
for  a  place  whose  longitude  is  60°  west,  the  time  by  an  ordinary 
clock  on  that  same  day,  when  the  time  by  an  astronomical  clock  at 
the  place  was  14  hours.  (Degree,  T.  C.  D.,  Hilary,  1893). 

Here,  local  sidereal  time  =14  sidereal  hours ; 
.'.     Greenwich  „          =14  +  ff=18  sidereal  hours. 

But  mean  time  =  sidereal  time  -  R.  A.  mean  sun 

=  18-10  =  8  sidereal  hours. 

But  8  sidereal  hours  =  7h  58m  419  mean  time  (Art.  160) ; 
.-.     Greenwich  mean  time  =  7h  58m  41s  p.m. 

From  this  must  be  subtracted  lm  18s  (approximately)  due  to  the 
change  in  the  mean  sun's  right  ascension. 

.-.     The  corrected  Greenwich  time  =  7h  57m  23§. 

But  difference  between  Greenwich  and  local  mean  times  =  4 
mean  solar  hours. 

.-.     Local  mean  time  =  3h  57m  23». 
o  2 


196 


TIME. 


[CHAP.  xi. 


To  find  at  what  time  a  Star  will  cross  the  Meridian. 

163.  Let  x  (fig.  77)  represent  a  star  in  the  meridian  AQ. 
Then  m  being  the  mean  sun,  we  have — 

T  Q  =  E.  A.  of  star, 
Tm  =  E.  A.  mean  sun, 
Qm  =  mean  time  of  star's  transit  (in  arc). 
But  Qm  =  TQ  -  °cm. 

A 


Q 


FIG.  77. 

Therefore,  we  have  the  following  equation,  all  the  quan- 
tities being  supposed  reduced  to  the  same  units — 
(Meantime  of  transit)  =  (E.  A.  star)  -  (E.  A.  mean  sun). 

EXAMPLE. — Find  at  what  time  a  Aquilse  will  cross  the  meridian  of 
Greenwich,  being  given  the  right  ascension  of  the  star  =  19h  43m  51", 
and  the  right  ascension  of  the  mean  sun  at  Greenwich  mean  noon 
=  Oh6m4O. 

H.         M.       8. 

E.  A.  star  =  19     43     51 

E,.  A.  mean  sun    =    0       6     40 


.'.     Mean  time  transit  =  19     37     11  (reduced). 
Reducing  19h  37m  11s  to  mean  time  (Art.  160),  we  get— 

H.       M.        S. 

Mean  time  of  transit  =19     33     58  after  mean  noon 

=    7     33     58  a.m.,  or7h30m44»  a.m., 

after  an  approximate  correction  for  the  change  in  the  K.A.  of  the 
mean  sun. 


€HAP.  XI.]       EQUINOCTIAL  TIME.       THE  CALENDAR.  197 

N.B. — When  the  transit  is  across  some  other  meridian 
than  that  of  Greenwich,  a  correction  must  be  made  for  the 
ohange  in  the  position  of  the  mean  sun  corresponding  to  the 
difference  of  longitude  of  the  two  places.  This  correction  is 
made  in  the  same  way  as  in  Art.  162. 

Equinoctial  Time. 

164.  In  addition  to  the  apparent,  mean  solar,  and  sidereal 
times,  another  kind  of  time  is  sometimes  used  which  is  inde- 
pendent of  the  position  of  the  observer  on  the  earth. 

The  Equinoctial  Time  at  any  instant  is  the  interval,  mea- 
sured in  mean  solar  days,  hours,  &c.,  since  the  preceding 
vernal  equinox. 

The  Calendar. 

165.  The  ordinary  civil  year  contains  an  exact  number 
of  days,  viz.  365.     But  the  time  taken  by  the  sun  to  com- 
plete a  revolution  in  the  ecliptic  is  about  365|  days.     The 
exact  interval  between  two  successive  vernal  equinoxes  is 
365d  5h  48m  45  5s.     This  period  is  called  a  Tropical  Year. 
A  Sidereal  Year,  or  the  time  taken  by  the  sun  to  return  to 
the  same  position  relative  to  the  fixed  stars,  is  slightly  longer 
than  the  tropical  year  on  account  of  the  precession  of  the 
equinoxes. 

Thus,  by  taking  the  civil  year  as  365  days,  there  is  an 
error  compared  with  the  tropical  year  of  5h  48m  45* 5s,  which 
in  four  years  amounts  to  23h  15m  2s,  or  very  nearly  a  day. 
If  this  error  were  not  corrected,  the  result  would  be  that  the 
dates  of  the  equinoxes  and  solstices  would  be  later  by  one 
day  every  four  years. 

166.  The  first  exact  attempt  at  approximating  the  length 
of  the  civil  to  that  of  the  tropical  year  was  made  in  the  time 
of  Julius  Caesar.     It  was  then  agreed  that  an  additional  day 
should  be  given  to  every  fourth  year,  which  was  to  contain 
366  days.     Such  a  year  is  called  a  bissextile  or  leap  year. 


198  TIME.  [CHAP.  xi. 

Those  years  are  chosen  as  leap  years,  which  are  divisible  by 
4  without  remainder,  such  as  1888,  1892,  &c. 

The  Calendar  constructed  in  this  manner  is  called  the 
Julian  Calendar. 

According  to  the  Julian  Calendar  a  correction  is  made  of 
one  day  every  four  years.  But  one  day,  or  24  hours,  is  in 
excess  of  23h  15m  2s  by  about  45  minutes.  Thus  the  correc- 
tion by  means  of  leap  year  leads  to  a  new  but  very  much 
smaller  error  of  about  45  minutes  in  four  years,  or  an  aver- 
age of  rather  more  than  11  minutes  each  year.  This  error, 
in  400  years,  would  amount  to  nearly  three  days. 

Hence  we  have  the  Gregorian  correction  to  the  Julian 
Calendar  adopted  by  Pope  Gregory  XIII.  in  1582,  according 
to  which  each  year  which  is  a  multiple  of  100,  such  as  1700, 
1800,  1900,  which,  by  the  Julian  Calendar,  are  leap  years, 
should  be  ordinary  years,  with  the  exception  of  those  years 
in  which  the  number  of  the  century  is  divisible  by  4  without 
remainder,  such  as  2000,  2400,  which  should  remain  leap 
years.  This  arrangement  evidently  makes  the  required  cor- 
rection of  three  days  in  400  years. 

Even  with  the  Gregorian  correction  there  is  still  a  very 
small  error,  which,  however,  would  amount  to  not  more 
than  a  day  in  20,000  years.  The  Gregorian  correction  was 
not  adopted  in  England  until  the  year  1752,  when  the 
accumulated  error,  as  compared  with  the  corrected  calendar, 
amounted  to  eleven  days.  Eleven  days  of  that  year  were 
therefore  skipped,  the  2nd  of  September  being  called  the 
13th. 

In  Russia,  where  the  Julian  Calendar  is  still  adhered 
to,  the  dates  are  thirteen  days  behind  those  of  the  rest  of 
Europe. 

The  Sun-dial. 

167.  In  a  sun-dial  the  apparent  time  is  indicated  by 
means  of  the  shadow  cast  by  a  rod  of  metal  on  a  horizontal 


CHAP.  XI.]  GRADUATION  OF  A  SUN-DIAL.  199 

plane.  The  rod  is  called  a  gnomon  or  stile.  The  gnomon 
points  to  the  celestial  pole,  and,  therefore,  makes  an  angle 
with  the  horizontal  dial-plate  equal  to  the  latitude  of  the 
place. 

The  principle  on  which  the  sun-dial  is  constructed  can 
be  easily  understood  by  supposing  the  observer  situated  at  0 
(fig.  78),  the  centre  of  a  sphere,  ON  being  the  direction  of 
the  celestial  pole.  If  12  equidistant  great  circles  be  drawn 
through  JVand  S,  the  sun  H  will,  in  his  diurnal  motion,  be  in 
the  plane  of  one  of  these  circles  at  the  termination  of  each  hour, 


FIG.  78. 

and  the  shadow  cast  by  a  gnomon  fixed  in  the  direction  ON 
will  coincide  each  hour  with  one  of  the  numbers  1,  2,  3,  &c., 
which  mark  the  intersection  of  the  hour-circles  with  the 
horizontal  circle  AB.  A  horizontal  sun-dial  is,  therefore, 
graduated  into  intervals  proportional  to  the  spaces  between 
these  intersections.  Hence  these  graduations  on  a  dial-plate 
which  mark  the  hours  are  not  generally  at  equal  intervals 
apart,  as  they  would  be  if  the  plane  of  AB  were  at  right 
angles  to  NS.  The  24  hour-angles,  however,  subtended  at 
the  celestial  pole  N  are  all  equal. 


200  TIME.  [CHAP,  xi 

Some  sun-dials  are  constructed  with  the  dial-plate  vertical 
instead  of  horizontal.  In  this  case  the  graduations  corre- 
spond to  the  intersections  of  a  vertical  circle,  whose  centre  is 
0,  with  the  hour-circles  passing  through  N  and  8. 


EXAMPLES. 

1.  The  times  of  the  sun's  rising  and  setting  on  November  1st  are  6h  56m 
and  4h  32m,  respectively  ;  find  approximately  the  equation  of  time. 

Am.  -  16m. 

2.  Convert  22h  26™  1s  sidereal  time  into  mean  solar  time  for  the  meridian 
of  Greenwich,  being  given  the  R.  A.  of  mean  sun  at  mean  noon  as  24h  4m  17s. 

Am.  2h  21™  21s. 

3.  In  question  2,  convert  2h  26m  12s  mean  solar  into  sidereal  time  for  the 
same  meridian. 

Am.  20h  30m  53s. 

4.  At  New  York,  in  longitude  74°  10'  "W.,  an  observation  is  made  on  August 
25th,  1893,  at  6h  3m  4s  mean  solar  time;  find  the  corresponding  sidereal  time, 
being  given  from  the  "  Nautical  Almanac  "  that  the  sidereal  time  of  mean  noon 
(R.  A.  mean  sun  at  noon)  at  Greenwich  on  above  date  is  10h  15m  54s. 

Ana.  16h  20m  46S'3. 

5.  Mars  revolves  on  its  axis  in  24h  37m,  and  round  the  sun  in  686  days  ; 
find  by  how  much  the  mean  solar  day  on  Mars  exceeds  that  of  the  sidereal  day. 

Ans.     2m  9s. 

6.  Find  the  R.  A.  of  the  true  sun  at  true  noon  on  November  25th,  1893, 
being  given  the  following  from  the  "  Nautical  Almanac  "  for  1893  :  — 

Equation  of  time  at  mean  noon  Nov.  25th  =  —  12m  45s. 
Sidereal  time  of  mean  noon  on  Sept.  2nd  =  10h  47m  26s. 

Here  the  R.  A.  of  mean  sun  at  mean  noon  on  September  2nd  =  10h  47m  26s. 
But  increase  in  the  R.  A.  from  September  2nd  to  November  25th 

=  24  x  ™  =  5h  31m  10s. 


R.  A.  of  mean  sun  at  mean  noon  on  November  25th  =  16h  18m  36s. 

But  change  in  R.  A.  in  12m  45s  =  2s  ; 
.-.     R.  A.  of  mean  sun  at  true  noon  =  16h  18ra  34s. 
But  true  sun's  R.  A.  —  mean  sun's  R.  A.  =  equation  of  time 
.-.    true  sun's  R.  A.  -  16h  18™  34s  =  -  12™  24s  ; 
.-.     true  sun's  R.  A.  at  true  noon  =  16h  6™  108. 


CHAP.  XI.]  EXAMPLES    ON    TIME.  201 

7.  (1)  The  maximum  values  of  the  equation  of  time  due  to  the  obliquity  of 
the  ecliptic  being  ±10  minutes,  and  due  to  the  eccentricity  of  the  earth's  orbit 
t  7  minutes,  show  that  the  equation  of  time  vanishes  four  times  a  year. 

(2)  How  many  times  a  year  would  it  vanish  were  the  magnitudes  of  tLe 
maxima  reversed? 

Am.     (1)  see  Art.  (156). 
(2)  twice. 

8.  Find  the  mean  solar  time  at  Madras,  longitude  80°  14'  19"  E.,  corre- 
sponding to  apparent  time  8  p.m.  there  on  September  6th,  1893,  being  given 
the  following  from  the  "  Nautical  Almanac  "  for  1893  : — 

At  Greenwich,  mean  noon. 

Equation  of  time  on  September  6th  =  -  1™  52"22. 
7th  =  - 2- 12-42. 

Ans.     7h  58m  5».54. 

9.  The  longitude  of  Dublin  being  6°  40'  W.,  and  that  of  Paris  2°  20'  E.; 
find  the  time  at  Paris  when  it  is  11.30  a.m.  at  Dublin. 

Ans.     12h  6m. 

10.  The  longitude  of  Pulkowa  being  30°  19'  40"  E.,  and  that  of  New  York 
74°  1'  W. ;  find  the  time  at  New  York  when  i  is  3.30  p.m.  ai  Pulkowa. 

Ans.     8h  32m  37s  a.m. 


(    202    ) 


CHAPTER  XII. 

APPLICATION   TO    NAVIGATION. 

168.  By  observing  the  heavenly  bodies,  we  are  enaoled 
to  determine  the  latitude  and  longitude  of  a  place  on  the 
earth  or  of  a  ship  at  sea.     But  the  instruments  used  in  a  fixed 
observatory  would  be  altogether  useless  at  sea,  owing  to  the 
motion  of  the  ship.      For  the  same  reason,  the   artificial 
horizon  we  avail  ourselves  of  on  land,  viz.  the  surface  of  a 
vessel  of  mercury,  would  not  at  sea  remain  a  horizontal 
plane.      What  is  required  in  such  observations  as  taking  the 
altitude  of  a  heavenly  body,  or  finding  the  angular  distance 
between  two  bodies,  is  an  instrument  by  means  of  which  the 
measurement  can  be  made  by  observing  both  objects  at  once, 
and  not  by  two  successive  adjustments  to  each  object,   of 
which  the  instability  of  the  ship  would  not  allow.     Such 
measurements,  unaffected  by  the  motion  of  the  ship,  can  be 
made  with  Hadley's  Sextant. 

Hadley's  Sextant. 

169.  This  instrument   consists   of  a  fixed    framework, 
formed  by  a  graduated  arc  AB  (fig.  79),  and  two  fixed  arms 
AM  and  JBM,  M  being  situated  at  the  centre  of  the  circle 
formed  by  AB.     Another  arm  VM  turns  round  the  centre 
M,  its  other  extremity    V  travelling  along  the  graduated 
arc  AB.     This  movable  arm  carries  a  small  mirror  at  M, 
called  the  index  glass,  which  moves  with  the  arm,  its  plane 
being  perpendicular  to  the  plane  of  the  instrument.     On  one 
of  the  fixed  arms  BM  is  a  fixed  mirror  N  called  the  horizon 
glass,  whose  plane  is  parallel  to  the  other  fixed  arm  AM,  so 


CHAP.  XII.] 


HADLEY  S    SEXTANT. 


203 


T 


that  when  the  movable  arm  VM  coincides  with  AM  the  two 
mirrors  are  parallel*  that 
is,  their  angle  of  inclina- 
tion is  zero.  The  point  A 
of  the  graduated  arc  is, 
for  this  reason,  the  zero 
point  of  the  scale.  Half 
of  the  horizon  glass  is  sil- 
vered, so  as  to  act  as  a 
mirror,  the  other  half  being 
plain  glass,  and  therefore 
transparent.  A  small  tele- 
scope T  is  fixed  to  the 
arm  AM,  and  directed  to 
the  horizon  glass. 

170.  Let  us  now  see  how,  by  means  of  this  instrument, 
the  angular  distance  between  two  objects  such  as  H  and  8 
(fig.  79)  can  be  found.  The  sextant  is  held  so  that  its 
plane  passes  through  both  objects,  and  in  such  a  position 
that,  in  looking  through  the  telescope,  one  of  the  objects  H 
can  be  seen  through  the  unsilvered  half  of  the  mirror  N. 
The  mirror  M  is  then  rotated  by  means  of  the  arm  VM  until 
the  image  of  8  coincides  with  that  of  H.  In  this  case  the 
rays  from  8  are  doubly  reflected,  first  from  the  index  glass 
and  then  from  the  horfzon  glass,  along  the  lines  SMNT. 
Having  got  the  images  of  S  and  H  coincident,  the  arm  VM 
is  clamped,  and  the  arc  A  V  is  read  off  by  means  of  a  vernier 
at  V.  This  reading,  on  being  doubled,  gives  the  angular 
distance  between  the  objects.  Usually,  however,  the  half 
degrees  in  the  arc  are  numbered  as  whole  ones,  so  that  the 
reading  of  the  vernier  at  once  gives  the  required  distance. 

The  arc  AV  (fig.  79)  evidently  measures  the  angle  at 
which  the  mirrors  are  inclined  to  one  another,  for  it  measures 
the  angle  between  AM  and  the  mirror  at  Mt  AM  being 
parallel  to  the  mirror  N. 


204 


APPLICATION  TO  NAVIGATION. 


[CHAP.  xn. 


171.  The  principle  on  which  Hadley's  Sextant  is  con- 
structed is  that  the  angular  distance  between  two  distant 
bodies,  such  as  S  and  Hy  is  double  the  angle  between  the 
planes  of  the  mirrors  when  the  image  of  S,  after  double 
reflection,  is  made  to  coincide  with  that  of  H.  To  prove 
this :  the  ray  SM  (fig.  80)  is  reflected  from  the  mirror  M  in 
the  direction  MN9  the  incident  and  reflected  rays  making 


FIG.  80. 

equal  angles  with  M\  let  each  of  these  angles  be  a.  Again, 
the  ray  MN  is  reflected  from  the  horizon  glass  along  the 
line  OH  (since  the  two  images  are  coincident).  Let  the 
angles  made  with  the  mirror  N  be  each  called  /3.  Now  we 
have  (Euclid,  i.  32)  — 

The  external  angle  2/3  =  2a  +  0  (fig.  80)  ; 


For  the  same  reason  — 

/3  =  a  +  0, 

.-.    0  =  /3-a; 
.-.       =  20. 


CHAP.  XII.]  THE  CHRONOMETER.  205 

But  0  is  the  angular  distance  of  S  from  H  as  observed  from 
0,  and  6  is  the  angle  between  the  planes  of  the  mirrors. 

172.  The    principal    use    of  Hadley's    Sextant  is    for 
measuring  the  altitude  of  the  sun.     The  observer,  holding 
the  instrument  upright,  looks  through  the  unsilvered  portion 
of  the  horizon  glass  at  that  part  of  the  horizon  which  is 
vertically  beneath  the  sun;  he   then  rotates  the  movable 
arm,  and  with  it  the  index  glass,  until  the  lower  edge  of  the 
sun's  image  just  touches  the  horizon.     Then  the  reading  of 
the  vernier,  after  correction  for  refraction,  "  dip "  of  the 
horizon,  and  other  errors,  gives  the  altitude  of  the  lower 
limb  of  the  sun,  to  which  the  sun's  semi-diameter  will  have 
to  be  added,  in  order  to  determine  the  altitude  of  the  sun's 
centre. 

The  instrument  is  called  a  sextant,  because  the  arc  AB 
(fig.  79)  generally  contains  60°.  Angular  distances  up  to 
120°  can  therefore  be  measured. 

The  Chronometer. 

173.  Every  ship  carries  one  or  more  very  accurately  con- 
structed watches,  called  chronometers,  which,  before  leaving 
port,  are  set  to  Greenwich  time.     As  it  is  of  the  greatest 
importance  that  Greenwich  time  should  be  accurately  known 
during  the  voyage,   it  is  necessary  that   the   chronometer 
should  go  at  a  rate  as  nearly  uniform  as  possible.     It  is  not 
necessary  that  it  should  keep  correct  time,  but  only  that  the 
amount  of  gain  or  loss  should  be  the  same  from  day  to  day. 
So  that  this  "  rate  "  being  known,  the  correct  Greenwich 
time  can  be  calculated  by  allowing  for  the  total  error  accu- 
mulated since  it  was  correctly  set. 

174.  Chronometers  differ  from  ordinary  watches  in  two 
particulars— (1)    the   peculiar   construction   of  the  balance 
wheel,  so  as  not  to  be  affected  by  changes  of  temperature ; 
(2)  the  "  detached  escapement."     If  the  balance  wheel  were 


206  APPLICATION  TO  NAVIGATION.  [CHAP.  XII. 

to  consist  of  an  entire  circle,  composed  of  one  metal 
alone,  then  an  increase  of  temperature  would  cause  it 
to  expand,  and  the  time  of  oscillation  would  be  increased, 
causing  the  watch  to  go  more  slowly.  However,  the 
circumference  of  the  wheel  is  generally  composed  of  three 
unconnected  arcs  :  the  external  portion  of  each  arc  is  brass, 
and  the  internal  part,  steel.  When  the  temperature  rises,  the 
brass  expands  more  than  the  steel,  causing  the  extremities  of 
the  arcs  to  curve  inwards  towards  the  centre,  thus  compen- 
sating for  the  expansion  of  the  spokes,  which  cause  the  arcs 
to  be  pushed  outwards.  Each  arc  is  also,  for  purposes  of 
adjustment,  weighted  with  little  screws.  The  "  detached 
escapement "  is  an  arrangement  by  which  the  action  of  the 
main  spring,  which  keeps  up  the  motion,  is  suspended  for 
the  greater  part  of  the  oscillation,  so  that  the  isochronism 
of  the  balance  wheel  is  hardly  at  all  affected  by  external 
impressions. 

Latitude  at  Sea.     Meridian  Observations. 

175.  First  Method.  The  latitude  at  sea  can  be  found  by 
taking  the  meridian  altitude  of  the  sun  with  the  sextant.  The 
observations  commence  some  time  before  apparent  noon  ;  and 
the  altitude  is  repeatedly  taken  until  it  ceases  to  increase,  and 
thus  the  maximum  or  meridian  altitude  is  found.  Also  the 
sun's  declination  is  given  for  each  day  at  Greenwich  noon  in 
the  "  Nautical  Almanac,"  together  with  its  rate  of  variation 
per  hour.  Therefore,  after  correcting  for  this  change  in 
declination,  due  to  the  interval  between  Greenwich  and  local 
noon  (Greenwich  time  being  given  by  the  ship's  chronometers), 
we  get  the  sun's  declination  at  that  instant.  The  latitude  is 
now  given  by  the  formula  : — 

(colat.)  ±  (declination)  =  (meridian  alt.).  (Art.  34.) 

In  our  latitudes  the  plus  or  minus  sign  is  taken  according 
as  the  declination  of  the  sun  is  north  or  south. 


CHAP.  XII.]  LATITUDE  AT  SEA.  207 

Similarly,  the  latitude  may  be  found  by  observing  the 
meridian  altitude  of  a  star  (or  other  body)  whose  declination 
is  known,  the  same  formula  colat.  +  S  =  a,  being  used.  If  the 
star  cross  the  meridian  between  the  zenith  and  the  pole  the 
formula  becomes  colat.  +  S  =  180  -  a  (Art.  34) ;  and  should  the 
altitude  be  observed  when  the  star  crosses  the  meridian 
between  P  and  R  (fig.  20),  it  changes  to  -  colat.  +  S  =  a,  the 
sign  of  8  being  in  all  cases  altered  when  it  represents  a  south 
declination. 

It  would  be  better,  however,  in  numerical  examples,  for 
the  student,  in  each  case,  instead  of  relying  on  the  formula, 
to  draw  a  diagram  and  directly  deduce  the  result  after 
placing  the  star  in  its  proper  position  corresponding  to  the 
given  measurements. 

EXAMPLE — The  sun's  meridian  altitude  on  December  4th,  1893, 
is  observed  to  be  16°  8'.  The  chronometer  indicates  6h  5m  12' 
Greenwich  time.  The  "Nautical  Almanac"  gives  the  sun's  decli- 
nation at  the  preceding  noon  at  Greenwich  as  22°  19'  25"  south, 
and  his  hourly  change  in  declination  as  19"- 6.  Find  the  latitude 
of  the  ship. 

Here  we  have : — 

Sun's  decimation  at  Greenwich  noon  =  22°  19'  25"  S* 
Hourly  change  (increasing)    =  19"*6; 

.-.     Change  in  6h  5m  12s  V  59"-3 ; 

.-.     Declination  at  local  noon        =  22°  21'  24'''3. 
B  colat.  -  declination  =  meridian  altitude ; 

.•.     colat.  -  22°  21'  24"-3  =  16°  8' ; 
.-.     colat.  =  38°  29'  24"'3  ; 
.-.     latitude  =  51°  30'  35"-7. 

Ex-meridian  Observations. 

176.  Second  Method.  By  simultaneously  observing  the 
altitudes  of  two  known  stars. 

Let  S,  S'  be  the  two  stars  (fig.  81)  when  their  altitudes 
are  measured.  Join  Z  and  P  with  S  and  £'  by  arcs  of  great 
circles. 


208  APPLICATION  TO  NAVIGATION.  [CHAP  XII. 

In  order  to  find  the  latitude  it  is  necessary  to  solve  three 
spherical  triangles.  In  the  triangle  SPS'  the  polar  distances 
PS,  PS'  of  the  stars  are  known,  since  they  are  the  comple- 
ments of  their  declinations  which  are  given  in  the  "  Nautical 
Almanac."  Also  the  angle  SPS'  is  known,  since  it  measures 
the  difference  between  the  known  right  ascensions  of  the 
stars ;  hence  the  side  SS'  and  the  angle  PSS'  can  be 
calculated. 

Again,  in  the  triangle  ZSS'  the  zenith  distances  Z8t  ZS' 
are  known,  being  the  complements  of  the  observed  altitudes, 
also  the  base  SSf  is  known,  therefore  the  angle  ZSS'  can  be 
found,  and  hence  the  angle  ZSP. 


FIG.  81. 

Lastly,  in  the  triangle  ZSP  the  two  sides  ZS,  PS  are 
kuown,  and  the  included  angle  ZSP  :  therefore  the  colatitude 
ZP  can  be  found. 

177.  Third  Method.  The  latitude  may  also  be  found  by 
taking  two  altitudes  of  the  sun,  and  noting  the  interval  of 
time  beween  the  two  observations.  This  is  practically  the 
same  method  as  the  last,  for  the  interval  of  time  between  the 
two  observations,  reduced  to  degrees  at  the  rate  of  15°  to  each 
hour,  gives  the  value  of  the  angle  SPS'  when  the  solution 
of  the  different  spherical  triangles  enables  us  to  find  the 
latitude  as  before. 


CHAP.  XII.]  LATITUDE    AT    SKA.  209 

The  latitude  can  also  be  found  by  taking  a  single  altitude 
of  the  sun,  provided  the  local  time  be  known. 

For  if  8  (fig.  81)  be  the  position  of  the  sun,  we  have  in 
the  triangle  ZPS  the  two  sides,  ZS  and  SP  are  known.  Also 
the  angle  ZPS  is  known,  being  the  hour  angle  of  the  sun, 
which  measures  the  local  apparent  time.  Therefore,  the 
colatitude  ZP  can  be  calculated. 

To  find  Mean  Local  Time. 

178.  First  Method.     By  equal  altitudes.     The  mean  local 
time,  or,  as  it  is  called  at  sea,  the  ship  mean  time,  can  be 
calculated  as  follows :— Observe,  with  the  sextant,  the  altitude 
of  the  sun  some  time  before  it  crosses  the  meridian.     Again, 
after  its  transit,  note  the  instant  at  which  it  attains  the  same 
altitude  as  before.     The  mean  of  the  times  of  the  two  obser- 
vations (given  by  a  chronometer)  will  give  the  time  of  transit, 
that  is,  apparent  noon ;  from  which,  knowing  the  equation 
of  time  from  the  "  Nautical  Almanac,"  the  mean  time  can  be 
found. 

179.  Second  Method.     By  observing  the  attitude  of  a  known 
star,  or  of  the  sun,  moon,  or  a  planet  (when  the  body  is  in,  or 
near,  the  prime  vertical) . 

In  this  case  the  latitude  of  the  place  is  supposed  to  be 
known. 

Let  8  be  a  star  (fig.  81)  in,  or  near,  the  prime 
vertical  whose  declination  is  known  from  the  "Nautical 
Almanac,"  and  whose  altitude  is  measured.  Then  in  the 
triangle  ZSP  the  three  sides  are  known,  ZS  being  the 
complement  of  the  observed  altitude,  PS  the  complement  of 
the  declination  of  the  star,  and  ZP  being  the  colatitude ; 
hence  the  hour  angle  ZPS  of  the  star  can  be  found.  When 
this  is  reduced  to  time  (by  dividing  by  15)  and  the  result 
added  to,  or  subtracted  from,  the  known  right  ascension  of 
the  star  according  as  the  star  is  west  or  east  of  the  meridian, 
we  obtain  the  sidereal  time  of  the  observation,  which  can  be 


210  APPLICATION  TO  NAVIGATION.  [CHAP.  XII 

reduced  to  mean  solar  time  by  the  method  explained  in 
Chapter  XI. 

In  all  cases  when  the  sun,  moon,  or  a  planet  is  chosen 
instead  of  a  star,  the  altitude  of  the  lower  or  upper  limb  is 
measured,  and  to  this  is  added  the  semidiameter  of  the  body 
(obtained  from  the  "  Nautical  Almanac  ")  which  g  ives  the 
altitude  of  the  centre. 

The  reason  the  body  is  chosen  in,  or  near,  the  prime 
vertical  is  that,  in  that  position,  the  altitude  of  the  body  is 
most  rapidly  changing,  and  therefore  a  small  error  in  the 
observed  altitude  will  produce  the  least  possible  error  in  the 
calculated  time. 

This  latter  method  is  one  very  frequently  used  at  sea. 

Longitude  at  Sea 

180.  The  problem  of  finding  the  longitude  is  reduced  to 
finding,  as  accurately  as  possible,  the  Greenwich  time  corre- 
sponding to  the  ship  mean  time.  For  (Art.  159) — 

Longitude  (in  time)  =  Greenwich  mean  time  ~  ship  mean  time. 

There  are  two  methods  by  which  the  longitude  may  thus 
be  determined  : — 

(1)  By  the  chronometer,  and  (2)  by  lunar  distances. 

We  have  already  seen  how  Greenwich  time  is  given  by 
the  ship's  chronometers,  two  or  three  being  kept,  in  order  to 
check  one  another.  The  ship  mean  time  is  generally  found 
by  observing  a  star  in  the  prime  vertical  or  by  the  method  of 
equal  altitudes.  The  difference  between  the  two  times 
multiplied  by  15  gives  the  longitude  in  degrees. 

EXAMPLE. — On  April  6th,  when  the  sun's  altitude  was  first 
observed,  the  ship's  chronometer  indicated  10h  6m48;  and  again, 
when  the  altitude  was  the  same  as  in  the  first  observation,  the 
indication  was  4h  3m  12s.  Also,  it  was  known  that  the  chronometer 
gained  5  seconds  daily,  it  being  6  days  since  the  ship  left  port 
(the  chronometer  then  indicating  correct  Greenwich  time).  The 
equation  of  time  on  April  6th  was  2m  20s.  Find  the  ship's  longi- 
tude. 


CHAP.  XII.]  LONGITUDE  BY  LUNAR  DISTANCES.  211 

Here,  if  we  take  half  the  sum  of  the  two  chronometer  readings, 
12  hours  being  added  to  the  second  one,  we  get  13h  4m  38s,  or, 
subtracting  12  hours,  lh  4m  38s.  Prom  this  we  subtract  30  seconds, 
the  error  of  the  chronometer,  and  we  have — 

H.    M."  s. 

Greenwich  time  of  local  apparent  noon  =  1     4     8 
Equation  of  time  =  0     2  20 

.•.     Greenwich  time  of  local  mean  noon  =1     1  48 

Multiplying  by  15,  we  get  longitude  =  15°  27'  "W. 


Longitude  by  Lunar  Distances. 

181.  If  the  chronometers  on  board  a  ship  should,  from 
any  cause  go  astray,  so  as  not  to  be  available  for  indicating 
correct  Greenwich  time,  then  the  moon,  by  its  change  of 
position  with  reference  to  the  fixed  stars  in  its  motion  round 
the  earth,  serves  as  a  fairly  reliable  time-keeper.  In  fact,  we 
may  regard  the  whole  heavens  as  an  immense  clock-face,  the 
stars  as  dial  figures,  and  the  moon  as  a  moving  clock-hand. 

In  the  "  Nautical  Almanac  "  is  given  a  series  of  tables 
which  predict,  for  each  day,  the  distances  of  the  moon's 
centre  from  certain  bright  stars  or  planets  in  its  neighbour- 
hood for  every  three  hours  of  Greenwich  mean  time.  The 
observer,  whose  object  is  to  ascertain  Greenwich  mean  time, 
therefore  measures  with  the  sextant  the  distance  of  one  of 
the  given  stars  from  the  edge  of  the  moon's  limb ;  to  this 
must  be  added  or  subtracted  the  moon's  semi-diameter,  in 
order  to  find  the  distance  of  the  star  from  the  moon's  centre. 
The  tables  in  the  "  Nautical  Almanac  "  will  then,  on  being 
referred  to,  give  roughly  Greenwich  mean  time,  correspond- 
ing to  this  distance,  with  an  error  of  perhaps  one  or  two 
hours.  But  during  the  above  intervals  of  three  hours  we 
may  assume  that  the  angular  distance  of  the  moon  from  the 
star  changes  uniformly,  and  therefore,  by  a  statement  in 
proportion,  we  can  calculate  the  exact  Greenwich  mean  time, 
from  which,  knowing  the  local  time,  the  longitude  may  be 
found  as  before. 

P  2 


212  APPLICATION   TO   NAVIGATION.  [CHAP.  XII. 

182.  Clearing  the  Distance. — In  the  above  method,  by 
lunar  distances,  a  correction  has  to  be  made  for  refraction. 
Also  the  lunar  tables  are  calculated  to  predict  the  place  of 
the  moon  as  seen  from  the  centre  of  the  earth,  and  there- 
fore we  must  also  allow  for  parallax.     These  corrections,. 
which  are  somewhat  involved,    constitute  what  is    called 
"  clearing  the  distance." 

183.  The  moon's  motion  among  the  fixed  stars,  although 
very  much  faster  than  that  of  the  sun  or  planets,  is  not  suffi- 
ciently rapid  to  determine  by  the  above  method  the  longitude 
with  very  great  accuracy.     On  account  of  this  slowness  of 
movement,  a  small  error  in  the  observed  distance  will  produce 
a  comparatively  large  error  in  the  calculated  Greenwich  time, 
and  therefore  in  the  longitude.     If  the  moon  were  to  revolve 
about  the  earth  in  two  or  three  days,  it  would  be  possible 
to  find  the  longitude  as  easily  as  the  latitude. 

EXAMPLE. — On  January  2nd,  1893,  the  angular  distance  of 
Kegulus  from  the  moon's  centre  was  observed  to  be  44°  15',  the 
local  time  being  6h  30m  p.m. ;  at  Greenwich  at  3  p.m.  and  6  p.m., 
the  distances,  as  given  in  the  "Nautical  Almanac,"  were  45°v13'  19" 
and  43°  24'  48",  respectively.  Determine  the  longitude  of  the 
place. 

Here  we  have — 

Angular  distance  at  3  p.m.  =  45°  13'  19" 

„  at  time  of  observation  =  44°  15'    0" 


Change  in  interval  =    0°  58'  19"  =  3499" 

But  subtracting  the  angular  distance  at  6  p.m.  from  that  at  3  p.m., 
we  find  that  the  change  in  3  hours  is  1°  48'  31"  =  6511"; 

3499 


The  time  in  which  it  decreases  3499"  =  3h 


7 

Greenwich  time  =  3h  +  lh  36m  44s  =  4h  36m  44s. 

But  local  time  =  6h  30m ; 
.*.    Longitude  (in  time)  =  lh  53m  16s ; 
or,  multiplying  by  15,  =  28°  19'  E. 


•CHAP.  XII.]  LONGITUDE  BY  OTHER  METHODS.  213 

Another  method  of  determining  Greenwich  time,  and  there- 
fore the  longitude,  is  by  observing  the  occultation  of  a  star 
by  the  moon.  This  is  merely  a  modification  of  the  method 
by  lunar  distances. 

184.  Attempts  have  been  made  to  find  the  longitude  by 
means  of  the  eclipses  of  Jupiter's  satellites,  the  Greenwich 
time  at  which  the  eclipses  commence  being  foretold  in  the 
"  Nautical  Almanac."  However,  it  is  not  possible  to  pro- 
perly observe  the  eclipses  by  means  of  a  telescope  on  board 
ship  ;  and  even  on  land  it  is  difficult  to  tell  exactly  when  the 
eclipse  begins  or  ends.  Also  certain  other  celestial  signals 
-such  as  the  beginning  or  end  of  an  eclipse  or  the  bursting  of 
meteors  have  been  used  for  the  same  purpose. 

Since  the  invention  of  the  electric  telegraph,  the  longi- 
tude of  any  station  on  land  can  be  determined  very  readily 
by  signalling  the  local  time  at  any  instant  from  some  station 
with  which  it  is  in  telegraphic  communication,  and  whose 
longitude  is  known,  the  difference  of  the  longitudes  being 
found  by  multiplying  the  difference  of  the  times  by  15. 

EXAMPLES. 

1.  On  March  7th,  the  sun  was  observed  to  have  equal  altitudes  when  the 
chronometer  indicated  10h  54m  a.m.  and  8h  38m  p.m.  Greenwich  time  ;   hence 
calculate  the  longitude,  the  equation  of  time  at  Greenwich  noon  on  March  7th 
and  March  8th  being  11'  12"  and  10'  57",  respectively. 

Am.     53°42'W. 

2.  Given  the  sun's  computed  hour  angle  to  be  75°  E.  when  the  chronometer 
indicated  21h  9m  308.     Find  the  longitude,  the  equation  of  time  being  -  2m  10", 

Am.     32°55'W. 


214    ) 


CHAPTER  XIII. 

THE  FIXED  STARS;       SPECTRUM  ANALYSIS. 

185.  The  stars  are  suns  situated  at  such  immense  dis- 
tances from  the  earth  that  they  appear,  even  when  viewed 
through  the  most  powerful  telescopes,  as  mere  points  of  light. 
Each  of  these  distant  suns  is  in  all  probability  the  centre  of 
a  system  similar  to  our  own,  a  focus  from  which  light  and 
heat  are  distributed  to  bodies  of  the  same  nature  as  our 
planets,  the  motions  within  each  system   being  doubtless 
regulated  according  to  the  same  general  laws  which,  as  all 
our  observations  show,  hold  throughout  the  universe. 

We  have  seen  in  Chapter  VII.  •  that  in  attempting  to 
measure  the  annual  parallax  and  distances  of  these  stars 
our  efforts  to  thus  make  a  survey  of  the  heavens  are  in  the 
greater  number  of  instances  doomed  to  failure,  the  reason 
being  that  the  greatest  distance  available  to  us  as  a  basis  of 
observation,  viz.  the  diameter  of  the  earth's  orbit  (185,000,000 
miles)  actually  dwindles  down  to  our  conception  of  a  geo- 
metrical point  compared  with  the  vastness  io  which  it  has 
to  be  applied. 

In  the  present  chapter  the  classification  of  the  stars  i& 
dealt  with,  together  with  a  short  account  of  the  principal 
discoveries  which  have  been  made  in  modern  times,  chiefly 
by  means  of  spectroscopic  analysis,  into  their  nature  and 
physical  condition. 

186.  Star  magnitudes. — The  stars  are  classified  into 
different  "  magnitudes,"  according  to  their  degrees  of  bright- 
ness.    There  are  about  twenty  of  the  most  brilliant  stars- 


CHAP.  XIII.]  NUMBER  OF  THE  STARS.  215 

which  are  said  to  he  of  the  first  magnitude.  Of  these  there 
are  about  twelve  visible  to  observers  in  our  latitudes.  These, 
together  with  the  constellations  in  which  they  are  situated, 
are  as  follows  :— Sinus  (Canis  Major);  Aldebaran  (Taurus); 
Capella  (Auriga) ;  Vega  (Lyra);  Betelgeux  and  Bigel 
(Orion) ;  Procyon  (Canis  Minor) ;  Spica  (Virgo) ;  Begulus 
(Leo)  ;  Arcturus  (Bootes)  ;  Antares  (Scorpio)  ;  Altair 
(Aquila). 

187.  There  are  .also  some  stars  of  the  first  magnitude 
whose  south  declinations  are  so  great  that  they  are  only  visible 
to  observers  in  the  southern  hemisphere,  as  Canopus  (a  Argus) ;, 
a  and  j3  Centauri ;  a   Crucis ;  Achernar  (a  Eridani) ; ,  and 
TJ  Argus.     Those  of  the  second  magnitude  are  less  brilliant; 
there  are  about  50, of  these  visible  to  us.     An  example  of 
this  class  is  the  Pole  Star,     It  is  not  possible  with  the  naked 
eye  to   distinguish  those   beyond  the  6th  magnitude,  but 
further  divisions  of  telescopic  stars  down  to  the  7th,  8th,  9th,. 
and  still  lower  magnitudes  have  been  made.   These  divisions 
are  in  a  great  measure  arbitrary,  for  stars  belonging  to  the 
same  class  differ  considerably  in  brightness,  colour,  and,  as 
we  shall  presently  see,  in  their  physical  condition. 

188.  dumber  of  the  Stars.— The  number  of  the  stars 
on  the  whole  celestial  sphere  which  it  is  possible  to  distinguish 
with  the  naked  eye  is  about  6000 ;  and  from  any  one  place, 
even  on  a  favourable  night,  not  many  more  than  2000  are 
visible  at  the  same  time,  numbers  of  stars  near  the  horizon 
being   obscured   by  the   greater    thickness    of    atmosphere 
through  which  their  rays  have  to  penetrate.     This  number  is 
not  nearly  so  great  as  a  person  who  has  not  made  an  exact 
estimate  of  them  would  expect,  for  the  idea  conveyed  to  the 
mind  by  the  multitudes  of  shining  points  spangled  over  the 
heavens  is  that  they  are  innumerable.      With  the   aid   of 
telescopes  the  number  which  it  is  possible  to  see  amounts  to 
many  millions. 


216  FIXED  STARS.       SPECTRUM  ANALYSIS.       [CHAP.  XI 11. 

189.  The  Milky  Way. — On  any  dark,  clear  night  there 
will  be  seen  stretching  across  the  sky,  almost  in  a  great  circle, 
a  faintly  luminous  belt,  which  is  called  the  milky  way.     Its 
luminosity  varies  greatly  in  different  parts.     The  telescope 
shows  that  this  phenomenon  is  produced  by  the  light  from 
countless  multitudes  of  stars  which  cannot  be  individually 
distinguished  with  the  unaided  eye. 

190.  Star  Clusters. — In  certain  parts  of  the  heavens 
the  stars  seem  so  densely  packed  together  and  in  so  marked 
a  manner  as  to  lead  to  the  supposition  that  they  are  in  some 
way  connected.     Such  a  group  is  called  a  star  cluster.    The 
group  called  the  Pleiades  is  seen  with  the  naked  eye  to  con- 
sist of  about  six  stars,  but  with  the  aid  of  a  small  telescope 
the  number  is  increased  to  over  fifty.     We  have  another 
illustration  in  a  luminous  spot  in  Perseus  which  the  telescope 
reveals   as  consisting  of  great   numbers   of  stars  grouped 
together,  the  appearance  forming  a  most  beautiful  spectacle. 


FIG.  82. 
Star  Cluster  in  Hercules  (Sin  J.  HERSCHEL). 

191.  UTebnlce. — Besides  star  clusters  there  are  seen  with 
the  aid  of  a  telescope,  in  different  parts  of  the  heavens,  many 


CHAP.  XIII.]       NEBULJE.       PROPER  MOTIONS  OF  STARS.  217 

objects  which  appear  as  small  luminous  spots,  and  many  of 
which  cannot  be  considered  as  star  clusters,  as  they  do  not 
seem  to  be  resolvable  into  separate  stars.  These  are  called 
nebula.  Perhaps  the  most  remarkable  illustration  of  these 
objects  is  the  great  nebula  in  Orion,  which  appears  shining  as 
a  bluish  mass,  portions  of  which,  under  high  magnifying 
power,  are  seen  to  contain  numerous  stars. 

Another  remarkable  example  is  the  annular  nebula  in  the 
constellation  of  Lyra :  "  It  consists  of  a  luminous  ring  ;  but 
the  central  vacuity  is  not  quite  dark,  but  filled  in  with  faint 
nebula  like  a  gauze  stretched  over  a  hoop"  (Sir  Eobert 
Ball). 


FIG  83. 
Annular  Nebula  in  Lyra  (LORD  ROSSE). 


192.  Proper  Motions  of  Stars.— When  the  right 
ascension  and  declination  of  a  star  is  observed  over  a  long 
series  of  years  it  is  found,  after  allowing  for  changes  in  these 
quantities  due  to  precession,  nutation,  and  parallax,  that  its 
position  in  most  cases  is  slowly  changing  with  reference  to 
other  stars  in  its  neighbourhood.  Each  star  is  thus  said  to 
have  a  proper  motion  of  its  own  of  a  character  not  common 
to  the  other  stars.  The  name  "  fixed,"  therefore,  applied  to 
the  stars,  is  not  strictly  accurate,  and  may  only  be  used  to 
distinguish  them  from  the  rapidly  moving  planets. 


218  FIXED  STARS.       SPECTRUM  ANALYSIS.  [CHAP.  XIII. 

The  proper  motions  of  stars  are  due  partly  to  a  motion  of 
the  sun  with  the  whole  solar  system,  which  is  believed  to  be 
moving  through  space  towards  a  point  in  the  heavens  near 
A  Hercules,  thus  causing  those  stars  not  in  the  line  of  direc- 
tion of  motion  to  appear  displaced  in  the  opposite  direction. 
However,  making  allowance  for  displacements  due  to  this 
cause  there  is  no  doubt  that  the  stars  are  in  actual  motion 
themselves. 

193.  Double  Stars.— With  the  aid' of  powerful  tele- 
scopes it  is  seen  that  many  stars  which  otherwise  appear 
single  are  in  reality  double,  consisting  of  two  distinct  stars. 
Sometimes  the  two  components  are  nearly  of  the  same  mag- 
nitude, but  in  many  double  stars  they  are  unequal,  and  when 
this  is  the  case  they  are  often,  for  some  reason  not  as  yet 
explained,  of  different  colour,  the  smaller  having  a  tint 
higher  in  the  spectrum  than  the  larger :  for  instance,  if  the 
larger  be  reddish,  the  smaller  will  be  blue  or  green.  About 
10,000  of  these  double  stars  have  been  discovered.  Many 
stars  which  at  first  appear  double  are  only  apparently  so, 
owing  to  their  being  nearly  in  the  same  line  of  vision ;  they 
thus  appear,  when  projected  on  the  surface  of  the  celestial 
sphere,  very  close  together,  when  in  reality  they  are  far 
apart.  Among  the  most  remarkable  examples  of  double 
stars  are  Castor,  a  Herculis,  the  Pole  Star,  and  Sirius.  Some- 
times a  star  is  seen  to  consist  of  three  or  four  separate  com- 
ponents ;  a£  in  the  case  of  *,  Lyrse,  in  which  there  are  four 
stars,  three  white  and  one  red, 

.194.  Binary  Stars.— In  many  double  stars  the  two 
components  are  seen  to  be  in  motion,  each  describing  an 
ellipse  round  their  common  centre  of  gravity  as  focus,  which 
in  fact  is  a  consequence  of  the  law  of  universal  gravitation 
(Art.  70).  In  the  case  of  the  double  star  Castor  this  move- 
ment is  so  slow  that  many  centuries  will  elapse  before  each 


CHAP.  XIII.]        ORBITS  OF  BINARIES.      VARIABLE  STARS.  219 

will  have  made  a  complete  revolution.     Stars  connected  in 
this  manner  are  called  binary  stars. 

195.  Orbits    of    Binaries.  — The    apparent    angular 
magnitude  of  the  regular  orbit  of  binaries  round  one  another 
can  be  determined  by  .means  of  the  micrometer;  but  the 
dimensions  of  the  .orbit  in  miles  cannot  be  found  unless  the 
distance  of  .the  star,  or,  which  amounts  to  the  same  thing,  its 
annual  parallax  be  known.     In  the  case  of  a  Centauri  the 
parallax  is  about  .-75"  and  the  semi-axis  of  its  apparent  orbit 
has  been  estimated  at  about  17'5".     Hence  we  have,  by, 
taking  the  ratio  of  the  circular  measures  of  these  angles — 

17*5"  _  semi-axis  of  orbit  in  miles 
~W^=  92,000,000 

Knowing  the  dimensions  of  the  orbit  of  a  binary  star  and 
its  periodic  time,  it  is  possible  to  calculate  the  sum  of  the 
masses  of  the  two  components.  For  an  account  of  the 
method  used,  see  Art.  214. 

196.  Variable  Stars. — There   are    some    stars   whose 
brightness  is  not  constant,  to  which  the  name  variable  stars  is 
given.     Of  these  the  most  remarkable  are  those  which  are 
known  to  change  their  lustre  periodically.    It  will  be  suffi- 
cient here  to  mention  some  of  the  more  remarkable  examples 
of  these,  each  being  representative  of  a  distinct  class. 

197.  The   "Mira  Type."— The  star   o  Ceti   or  Mira 
(the  wonderful)  goes  through  a  regular  cycle  of  changes  in  a 
period  of  331  days,  during  which  time  it  varies  from  the  second 
to  the  sixth  magnitude  ;  it  then  becomes  invisible  for  about 
five  months,  after  ,which  it  gradually  returns  to  its  original 
brightness. ' 

198.  The  "Algol  Type."— The  star  Algol  in  the  con- 
stellation ^of  Perseus  is  a  type  of  another  remarkable  class  of 
periodical  stars.     It  remains  of  the   second  magnitude  for 


220  FIXED  STARS.       SPECTRUM  ANALYSIS.  [CHAP.  XIII. 

about  2d  13h  ;  its  brightness  then  gradually  diminishes  till  it 
appears  of  the  fourth  magnitude,  the  time  taken  being  about 
3J  hours.  For  twenty  minutes  it  remains  of  the  fourth  mag- 
nitude, when  it  gradually  recovers  its  brightness  until,  at  the 
end  of  another  3J  hours,  it  appears  again  of  the  second 
magnitude.  The  whole  period  of  this  cycle  of  changes  is 
2d  20h  48m  55s.  As  regards  stars  of  this  type,  the  most 
probable  explanation  is  that  the  loss  of  light  is  due  to  the 
interposition  of  some  dark  body  which,  revolving  in  a  fixed 
period  round  the  star,  causes,  at  regular  intervals,  a  partial 
eclipse  to  take  place. 

The  Spectroscope. 

199.  We  know  from  Optics  that  when  a  ray  of  ordinary 
white  sunlight  passes  through  a  prism  it  emerges  split  up 
into  component  rays  of  different  hues,  viz.  violet,  indigo, 
blue,  green,  yellow,  orange,  and  red  ;  so  that  when  thrown  by 
a  suitable  arrangement  on  a  screen  there  appears  a  distinct 
band  of  colours  like  a  rainbow,  having  violet  at  one  end  and 


FIG.  84. 

deep  red  at  the  other  (fig.  84).  This  band  constitutes  what 
is  called  the  spectrum  of  the  light.  In  a  similar  manner  the 
light  from  any  other  source  may  be  examined.  Usually  the 
rays  are  admitted  through  a  narrow  slit,  and,  before  reaching 
the  prism  are  passed  through  a  collimating  lens,  whose  focus 
coincides  with  the  slit,  from  which  they  therefore  emerge  as 
parallel  rays,  and,  after  passing  through  the  prism,  the 


CHAP.  XIII.]       REVELATIONS  OF  THE  SPECTROSCOPE.  221 

spectrum  is  viewed  through  a  telescope.     Such  an  arrange- 
ment is  called  a  spectroscope. 

200.  Thus  by  the  aid  of  a  simple  glass  prism  we  have 
the  means  of  analysing  the  light  from  any  source  by  decom- 
posing it  into  its  separate  parts,  and  arranging  these  parts 
before  our  view.     An  examination  into  any  deficiencies  or 
other  peculiarities  in  the  spectrum  of  a  beam  of  light  thus 
divided    enables   us   to   learn  a  great  deal  concerning  the 
composition  and  physical  state  of  the  luminous  body  from 
which  the  light  is  derived.     This  is  the  principle  of  the 
method  of  research  known  as  spectrum  analysis,  which  in 
recent  years  has  enabled  us  to  add  so  much  to  our  knowledge 
of  the  universe. 

201.  Solar  Spectrum. — Early  in  the  present  century 
it  was  discovered  by  Fraunhofer  that  the  spectrum  of  the 
sun  is  not  a  continuous  succession  of  colours,  but  is  inter- 
rupted by  thousands  of  fine  dark  lines.     In  certain  parts  of 
the  spectrum  but  few  of  these  dark  lines  occur,  but  in  other 
portions  they  are  so  crowded  together  that  it  is  difficult  to 
distinguish  them   individually.     It  is  also  found  that  the 
arrangement  of  these  lines  is  a  characteristic  and  permanent 
feature   of   sunlight,   so   characteristic    that   their   number 
and  the  positions  which  they  occupy  in  the  spectrum  enable 
us  to  distinguish   sunlight   from    that  due   to   any   other 
source. 

From  this  it  is  evident  that  only  those  rays  reach  us  from 
the  sun  which  are  of  certain  definite  degrees  of  refrangibility, 
while,  from  some  cause  unknown  at  the  time  of  Fraunhof er's 
discovery,  other  rays,  corresponding  to  the  dark  lines,  are 
absent.  This  phenomenon  was  at  last  explained  by  an  exami- 
nation of  different  kinds  of  artificial  light. 

202.  When    rays    of   light  from  different   sources   are 
observed   by  means   of   the   spectroscope  it   is   found   that 


222  FIXED  STARS.       SPECTRUM  ANALYSIS.          [CHAP.  XIII. 

their    spectra    may    be    divided    into    the    two    following 
classes: — 

(1)  The  spectra  of  luminous  solids  and  liquids  are  con- 
tinuous, containing  light  of  all  degrees  of  refrangibility,  and 
therefore  show  no  dark  lines. 

(2)  The  spectra  of  the   flames   of  burning   gases,   not 
containing  solid  particles  in  suspension,  are  discontinuous, 
consisting  merely  of  a  certain  number  of  finite  bright  lines 
interrupted  by  dark  bands. 

203.  Reversal  of  Bright  Lines. — If  a  burning  gas  or 
vapour  emitting,  as  we  now  see,  only  rays  of  certain  degrees 
of  refrangibility,  be  interposed  between  the  observer  and  the 
light  from  a  source  giving  a  continuous  spectrum,  the  gas 
will  absorb  rays  of  the  same  kind  as  those  which  it  emits,  and  it 
depends  on  the  relative  brightness  of  the  two  sources  of  light 
as  to  whether  this  particular  class  of  rays  shall,  in  the  com- 
bined spectrum,  be  darker  or  brighter  than  the  rest.  The 
following  experiment  will  serve  to  illustrate  this  important 
principle : — 

Let  the  spectrum  of  the  burning  vapour  of  sodium  (got 
by  colouring  the  flame  of  a  spirit  lamp  with  common  salt,  of 
which  sodium  is  a  constituent)  be  observed,  and  it  is  found 
to  merely  consist  of  two  bright  yellow  lines.  However,  if 
very  bright  limelight  be  placed  behind  the  sodium  flame,  it 
is  found  that  the  continuous  spectrum  which  limelight  would 
afford,  if  viewed  alone  (being  that  of  an  incandescent  solid], 
is  crossed  by  two  dark  lines  corresponding  to  the  two  bright 
lines  of  sodium.  On  removing  the  limelight  these  lines 
flash  out  as  bright  as  before.  The  bright  lines  produced  by 
sodium  are  thus  reversed,  i.e.  changed  to  dark  ones ;  not 
that  they  are  actually  darker  than  when  viewed  alone,  but 
that  they  appear  dark  by  contrast  with  the  brilliance  of  the 
rest  of  the  spectrum;  for,  those  particular  rays  from  the 
limelight  which  are  characteristic  of  sodium,  and  which 


CHAP.  XIII.]  SUKFACE  OF  THE  SUN.  223 

would  otherwise  illuminate  the  spaces  occupied  by  these 
lines,  so  as  to  make  them  appear  as  bright  as  the  rest  of  the 
spectrum,  are  cut  off  by  the  interposition  of  the  sodium  flame, 
while  all  other  rays  pass  freely  through. 

204.  By  means  of  the  above  principle,  the  occurrence  of 
dark  lines  in  the  solar  spectrum  can  now  be  explained  by 
supposing  the  sun  to  be  surrounded  by  an  external  vaporous 
layer  which  absorbs  rays  of  its  own  peculiar  kind  from  the 
light  coming  to  us  from  an  inner  stratum  of  the  sun  called 
the  photosphere,   which  would  otherwise  give  a  continuous 
spectrum.     KirchofJ  showed  that  these  lines  correspond  to 
those  of  hydrogen,  iron,   zinc,   nickel,    copper,    and   other 
metals  which,  existing  in  a  state  of  vapour  in  the  solar 
atmosphere,  cause  the  reversal  of  their  characteristic  rays. 

205.  Surface  of  the   Sun.     Solar  Prominences. — 

During  recent  total  eclipses  the  spectroscope  has  been  applied 
to  investigate  the  nature  of  the  outer  surface  of  the  sun. 
Ou.tside  the  photosphere  is  the  chromosphere,  so  called  on 
account  of  its  bright  red  colour,  due  probably  to  intensely 
heated  hydrogen,  of  which  it  is  mainly  composed.  Beyond 
this  lies  the  corona,  a  ring  of  light  surrounding  the  sun 
seen  during  a  total  eclipse,  whose  spectrum  gives  but  faint 
indications  of  hydrogen,  and  whose  chief  characteristic  is 
a  conspicuous  green  line. 

During  a  solar  eclipse,  when  the  sun's  disc  is  hidden  by 
the  moon,  there  are  seen,  apparently,  on  the  edge  of  the 
moon's  disc,  some  remarkable  objects  now  known  as  solar 
prominences.  At  first,  when  their  nature  was  unknown,  they 
were  simply  called  prominences,  as  it  was  uncertain  whether 
they  were  appearances  on  the  outer  surface  of  the  moon  or 
sun  ;  but  it  was  definitely  proved  that  they  were  solar  during 
the  eclipse  of  1860  by  means  of  photographs  which  showed 
that  the  moon's  disc  moved  over  them  just  as  it  did  over 


224  FIXED  STARS.       SPECTRUM  ANALYSIS.       [CHAP.  XIII, 

other  portions  of  the  sun's  surface.  They  appear  as  flame- 
like  objects,  scarlet  in  colour,  and  they  vary  in  a  wonderful 
manner  as  regards  their  form  and  magnitude.  Some  of 
them  attain  a  height  of  over  80,000  miles.  With  what 
rapidity  they  are  capable  of  changing  will  be  seen  from  an 
example  chronicled  by  Professor  Young,  and  observed  at 
Princeton,  New  Jersey,  in  which  one  of  these  prominences, 
which  at  first  was  about  40,000  miles  high,  suddenly  shot  up 
until  it  attained  an  elevation  of  350,000  miles,  when  it 
gradually  broke  up,  and  eventually  faded  completely  away, 
the  whole  series  of  changes  being  completed  in  an  interval 
of  two  hours.  These  prominences  are  now  known  to  be 
protuberant  portions  of  red  incandescent  gas,  principally 
hydrogen  from  the  chromosphere.  In  order  to  observe  these 
prominences  it  is  not  now  necessary  to  wait  until  a  total 
eclipse  of  the  sun  occurs,  as  by  a  proper  arrangement  of  the 
spectroscope  the  changes  in  their  structure  can  be  studied 
almost  as  well  as  during  an  eclipse,  with  the  additional  ad- 
vantage of  being  able  to  observe  them  for  a  much  longer 
period  of  time. 

206.  The  spectrum  of  a  planet  would  be  identical  with 
that  of  the  sun  were  it  not  that  it  is  somewhat  altered  owing 
to  the  solar  light  having  to  pass  twice  through  the  thickness 
of  the  planet's  atmosphere,  which  causes  absorption  of  some 
of  the  rays.     In  the  case  of  some  of  the  planets  there  are 
characteristic  lines  indicating  the  presence  of  vapour  of  water 
under  conditions   similar   to   that   which   is  present  in  the 
atmosphere   of  the   earth.     The   spectrum  of  the   moon  is 
identical  with  that  of  the  sun,  which  is  confirmatory  evidence 
that  it  possesses  no  atmosphere. 

207.  Star   Spectra. — The  stars  may  be   divided   into 
different  classes  according  to  their  spectra  : — 

(1)  Those  whose  spectra  are  distinguished  b}  compara- 
tively   few    lines,    the   most   prominent    corresponding    to 


CHAP.  XIII.]          SPECTRA  OF  STARS  AND  NEBULJE.  225 

hydrogen  at  a  very  high  temperature.  In  this  class  are 
included  all  the  white  or  bluish  stars  such  as  Sirius  and 
Vega. 

(2)  In  the  second  class  are  such  stars  as  Aldebaran  and 
Arcturus,  whose   spectra  are  similar  to   that   of  the   sun, 
i.e.   are  intersected  by  large  numbers  of  fine  lines  indicat- 
ing the  presence  of  many  metals  in  addition  to  hydrogen. 

(3)  In  the  third  class  are  those  whose  spectra  are  inter- 
rupted by  dark  broad  bands.     Most  of  these  stars  are  red, 
and  a  large  number  are  variable. 

Many  of  the  metals  present  in  the  sun  have  been  observed 
in  the  stars.  Thus  in  Aldebaran  there  is  evidence  of  sodium, 
iron,  bismuth,  antimony,  magnesium,  calcium,  mercury,  and 
tellurium. 

208.  Spectra  of  UTebulac. — It  was  first  observed  by 
Huggins  that  some  of  the  nebulae  afford  spectra  which  are 
not  continuous  bands  of  colours,  crossed  by  dark  lines,  like 
those  of  stars,  but  merely  consist  of  a  few  bright  lines.    Four 
bright  lines  are  usually  easily  observed,  two  of  which  are 
certainly  due   to  hydrogen,  but  the  nature  of  the   others 
remains  as  yet  unknown.     These  nebulae  cannot  therefore 
consist  of  aggregations  of  separate    stars,  but  of  glowing 
gaseous  material,  of  which  hydrogen  is  a  chief  constituent. 
The  great  nebula  in  Orion  is  typical  of  this  class.     Many 
other  nebulae,  however,  of  which  that  in  Andromeda  is  a  type, 
afford  spectra  of  a  totally  different  kind ;  instead  of  a  few 
bright  lines  we  find  a  continuous  spectrum,  which  would 
point  to  the  conclusion  that  the  light  may  be  due  to  an 
immense  cluster  of  minute  stars.     Nebulae  of  this  type  are 
generally  white,  while  the  gaseous  nebulae  are  of  a  bluish 
tint. 

209.  One   of  the  most  remarkable  applications  of  the 
spectroscope  is  to  measure  the  velocity  of  a  star  along  the 
line  of  sight  either  towards  or  from  the  earth.     If  rays  of 

Q 


226  FIXED  STARS.      SPECTRUM  ANALYST          [CHAP.  XIII. 

light  proceed  from  a  body  which  is  approaching  the  earth 
their  wave-lengths  will  be  diminished,  for  the  number  of 
vibrations  reaching  the  earth  in  each  second  will  be  greater 
than  if  the  body  were  at  rest.  On  the  other  hand,  when 
receding  from  the  earth  there  will  be  a  corresponding 
increase  in  the  wave-lengths.  But  the  refrangibility  of  each 
ray  depends  on  its  wave-length,  a  diminution  in  which 
causes  the  ray  to  fall  nearer  the  violet  end  of  the  spectrum. 
Hence,  by  comparing  the  positions  of  the  lines  in  the 
spectrum  of  a  star  corresponding  to  hydrogen  or  some  other 
well-known  substance  with  the  positions  which  these  lines 
occupy  in  independent  observations,  it  can  be  determined 
whether  the  wave-lengths  are  diminished  or  increased,  and 
consequently  whether  the  star  is  moving  towards  or  from  the 
earth,  and  with  what  velocity.  This  method  has  been 
applied  to  calculate  the  velocities  of  a  considerable  number 
of  the  stars,  and  the  results  obtained  in  some  cases  indicate 
a  speed  along  the  line  of  sight  of  from  20  to  30  miles  per 
second. 

"  The  theory  of  this  method  is  beautifully  verified  by 
observations  on  the  sun.  As  the  eastern  edge  of  the  sun 
is  approaching  and  the  western  is  receding,  there  is  a  cor- 
responding difference  in  the  spectra  of  the  two  edges,  and 
the  observed  amount  gives  a  velocity  of  rotation  practically 
coincident  with  that  otherwise  known." — (SiR  EGBERT  BALL.) 


(    227    ) 


CHAPTER   XIV. 

MASSES   OF   THE   HEAVENLY   BODIES. 

210.  The  mass  of  the  earth  can  be  found  by  comparing 
its  attractive   force   with  that   of  a  body  whose  mass   is 
accurately  known. 

211.  Maskelyne's  Method. — A  mountain  in  Scotland 
named  Schehallien  was  chosen  whose  shape  was  so  regular 
that  its  volume  could  be  accurately  ascertained.      Its  mean 
specific  gravity  was  also  calculated  from  a  knowledge  of  the 
proportions  in  which  the  different  minerals  to  be  found  in  the 
mountain  entered  into  its  composition.      Two  observatories 
were  then  placed,  one  on  the  north  and  the  other  on  the 
south  side  of  the  mountain,  in  the  same  meridian.      At  each 
of  these  stations  the  meridian  zenith  distance  of  some  chosen 
star  was  observed,  the  zenith  in  each  instance  being  deter- 
mined by  the  direction  of  a  plumb-line.      If  the  plumb-line 
during  each  observation  pointed  exactly  to  the  zenith  of  the 
place,  the  difference  in  the  two  observed  meridian  zenith 
distances  of  the  star  would  equal  the  difference  in  the  latitudes 
of  the  two  observatories.      It  was  found,  however,  that  the 
former  exceeded  the  latter  by  11"'6,  which  was  accounted 
for  by  the  attraction  exerted  by  the  mountain  on  the  plumb- 
lines,  causing  them  to  deviate,  the  one  slightly  to  the  north, 
and  the  other  to  the  south,  of  the  true  vertical,  the  sum  of 
the  deviations  being  11" '6. 

Again,   an   independent  calculation  was  made,  from   a 
comparison  of  the  volume  of  the  mountain  with  that  of  the 

Q2 


228  MASSES   OF   THE   HEAVENLY   BODIES.         [CHAP.    XIV. 

earth,  as  to  what  should  be  the  deviations  of  the  plumb-lines, 
assuming  that  the  mean  specific  gravity  of  the  earth  was  equal  to 
that  of  the  mountain,  and  it  was  found  that,  upon  this  suppo- 
sition, the  total  deviation  should  be  20"'8  instead  of  ll"-6. 
It  was  therefore  concluded  that  the  specific  gravity  of  the 
earth  exceeded  that  of  the  mountain  in  the  ratio  of  20*8  to 
11*6.  Taking  the  mean  specific  gravity  of  Schehallien, 
which  was  found  to  consist  of  nearly  equal  parts  of  quartz, 
mica  schist,  and  crystalline  limestone,  as  2-8,  it  followed  that 
that  of  the  earth  was  5*02,  from  which,  knowing  its  volume, 
its  mass  was  calculated. 

The  Cavendish  Experiment. — Thft  method  was  first 
adopted  by  Cavendish  in  1798.  A  light  wooden  rod  was 
taken  with  two  small  balls  attached  at  its  extremities  and 
suspended  in  a  horizontal  position  by  a  very  fine  wire. 
When  in  a  state  of  rest  two  large  metal  balls  were  brought 
close  to  the  smaller  ones  and  on  opposite  sides  of  the  wooden 
rod  (in  order  that  their  attractions  might  deflect  the  rod  in 
the  same  direction).  A  deflection  is  thus  produced,  which  is 
resisted  by  the  elasticity  of  the  wire.  The  positions  of  the 
large  balls  are  then  changed  by  placing  each  on  that  side  of 
the  rod  opposite  to  that  which  it  previously  occupied,  and  an 
equal  deflection  is  produced  in  the  opposite  direction.  The 
angle  of  deflection  in  each  case  is,  therefore,  half  the  diffe- 
rence between  the  extreme  positions  of  the  balls  in  each  case. 
The  angle  of  deflection  being  0,  the  magnitude  of  the  couple 
tending  to  untwist  the  wire  will  be  /0,  where  f  is  a  constant 
called  the  torsional  rigidity  of  the  wire. 

The  value  of  /  can  be  found  by  observing  the  periodic 
time  of  the  oscillations  of  the  rod  just  after  the  large  balls 
have  been  removed.  Hence  the  attractions  of  the  metal 
balls  on  the  smaller  ones  can  be  determined.  On  compari- 
son with  the  force  with  which  the  earth  attracts  the  small 
balls  the  ratio  of  the  mass  of  the  earth  to  that  of  one  of  the 
large  metal  balls  can  be  determined. 


CHAP.  XIV.]  MASSES  OF  THE  SUN  AND  PLANETS.  229 

Another  method,  first  adopted  by  Cavendish,  consists  in 
observing  the  attraction  of  tides  in  estuaries.  The  same 
object  can  also  be  effected  by  comparing  the  vibrations  of  a 
pendulum  at  the  top  and  bottom  of  a  deep  mine,  and  also  by 
several  other  experiments,  into  the  details  of  which  it  is  not 
here  advisable  to  enter. 

To  find  the  Ratio  of  the  Sun's  Mass  to  that  of  the  Earth. 

212.  Let  -R  denote  the  distance  of  the  earth  from  the 
sun,  and  T  its  periodic  time ;  also  let  r  and  t  denote  the 
moon's  distance  from  the  earth  and  periodic  time,  respectively. 
Therefore,  supposing  both  orbits  circular,  we  have — 

Earth's  centrifugal  force  =  — ™-  • 

™  47rV 

Moon's          „  „     =  — - 

But,  if  S  and  E  denote  the  masses  of  the  sun  and  earth 
respectively,  the  attraction  exerted  by  the  sun  on  the  earth  is 

Sf     J? 
to  that  of  the  earth  on  the  moon  as  j» :  —  (see  Newton's 

Law  of  Gravitation,  Art.  70).   Therefore  we  have  (neglecting 
in  each  case  the  attraction  of  the  smaller  body  on  the  larger) — • 

_£  fE      WR    47rV 
JB>  :  !*  :  :    T        tz  ' 

«:*::?:£ 

If  we  take  approximately  R  =  385r  and  T  =  13*,  the 
above  result  gives — 

S  =  ^E  =  337,672  E, 

and  consequently,  knowing  the  mass  of  the  earth,  that  of  the 
sun  can  be  calculated. 

213.  Masses  of  the  Planets.— Having  determined 
the  mass  of  the  sun,  that  of  any  planet  which  is  accompanied 


230  MASSES  OF  THE  HEAVENLY  BODIES.  [CHAP.  XIV. 

by  a  satellite  can  be  found.  Thus,  in  the  case  of  Jupiter,  let 
R  be  the  radius  of  its  orbit,  and  T  its  periodic  time,  while  r 
and  t  denote  the  orbital  radius  and  periodic  time  respectively 
of  any  one  of  its  satellites,  the  mass  J  of  Jupiter  can  be 
determined  by  the  proportion  — 

jR3    r3 
fy  .    T  .  .        . 

*>  •  J  '  '  ji  •  p' 

The  ratio  of  the  mass  of  the  sun  to  that  of  Jupiter  is  thus 
found  to  be  1047  to  1,  the  same  result  being  obtained  when 
the  calculation  is  based  upon  the  observed  periodic  time  and 
mean  distance  of  any  one  of  the  four  satellites. 

The  mass  of  a  planet  which  is  not  accompanied  by  a 
satellite  can  be  determined  by  the  perturbations  produced  by 
its  attraction  on  the  other  bodies  of  the  solar  system. 

214.  Masses  of  Binary  Stars.  —  If  we  know  the  semi- 
axis  major  of  the  orbit  of  a  binary  star  and  the  period  of 
revolution  (Art.  195),  we  can  calculate  the  ratio  of  the  mass 
of  the  sun  to  the  sum  of  the  masses  of  the  two  stars  by  'the 
proportion— 

S:M+M>:'/±:r-, 

where  r  is  the  semiaxis  major  of  the  orbit  of  the  binary,  and 
the  periodic  time,  M  +  M.'  being  the  sum  of  the  masses  of 
the  pair,  while  S,  J?,  and  T  denote  the  same  quantities  as  in 
the  previous  Article.  Thus,  in  the  case  of  the  binary  star 
a  Centauri,  the  semiaxis  major  of  its  orbit  is  found  to  be 
23*3  times  the  distance  of  the  earth  from  the  sun,  and  the 
period  of  revolution  77  years  ;  hence  in  this  instance  we 


or  Jf+Jf  =  2-14S, 

from  which  we  conclude  that  the  sum  of  the  masses  of  the 
two  stars  of  which  a  Cen'  uri  consists  is  more  than  twice 
that  of  the  sun. 


NOTE  ON  THE  CELESTIAL  GLOBE.  231 

NOTE  ON  THE  CELESTIAL  GLOBE. 

On  the  surface  of  a  celestial  globe  are  marked  the  appa- 
rent positions  of  the  stars  and  the  different  circles  of  the 
celestial  sphere.  The  globe  revolves  within  a  framework 
consisting  of  a  brass  meridian  (graduated),  which  remains  in 
a  vertical  plane,  and  a  broad  horizontal  wooden  ring  which 
represents  the  horizon.  On  this  wooden  horizon  are  marked 
the  months  and  days  of  the  year,  the  equation  of  time  for 
each  day,  and  the  daily  longitude  of  the  sun ;  there  are  also 
the  twelve  signs  of  the  zodiac  in  their  order,  and,  finally, 
a  circle  divided  into  degrees  for  the  purpose  of  measuring 
azimuths. 

The  bearings  on  which  the  globe  revolves  are  attached 
to  two  diametrically  opposite  points  of  the  brass  meridian 
corresponding  to  the  north  and  south  celestial  poles. 

At  the  northern  pole  of  the  globe  is  seen  a  small  brass 
circle  called  the  hour  index,  on  which  are  marked  the  hours  of 
the  day  as  on  the  face  of  a  clock.  This  circle  can  be  turned 
round  with  the  fingers,  so  that  any  hour  desired  may  coincide 
with  the  meridian,  but,  once  set,  there  is  sufficient  friction  to 
cause  it  to  turn  with  the  globe  when  the  latter  is  rotated. 

As  regards  the  different  circles  drawn  on  the  globe  it  is 
to  be  observed  that  the  equator  and  ecliptic  are  graduated 
into  degrees  and  fractions  of  a  degree. 

TO  set  the  Celestial  Globe  so  as  to  show  the  Appearance  of  the 
Heavens  at  a  given  place  and  at  any  given  apparent  time. 

(1)  Elevate  the  pole  to  an  angle  equal  to  the  latitude  of 
the  place  by  means  of  the  graduations  on  the  meridian. 

(2)  Find  the  position  of  the  sun  in  the  ecliptic  on  the 
day  in  question  (the  spot  can  be  marked  with  a  small  piece 
of  gum-paper  stuck  on  the  globe) ;  rotate  the  globe   until 
this  point  coincides  with  the  meridian,  and  set  the  hour  index 
at  XII.     This  position  corresponds  to  apparent  noon. 


232  NOTE  ON  THE  CELESTIAL  GLOBE. 

(3)  Finally,  turn  the  globe  until  the  required  hour  isr 
brought  to  coincide  with  the  meridian,  which  gives  the 
required  position. 

If  the  globe  be  set  so  that  the  plane  of  the  meridian,  is 
due  north  and  south,  the  actual  direction  as  well  as  the 
relative  position  of  any  star  in  the  heavens  will  be  indicated. 

To  determine  the  Apparent  Time  at  which  a  Heavenly  Body 
rises  or  sets  at  a  given  place. 

(1)  Make  the  elevation  of  the  pole  equal  to  the  latitude 
of  the  place. 

(2)  Eotate  the  globe  until  the  position  occupied  by  the 
sun  in  the  ecliptic  coincides  with  the  meridian,  setting  the 
hour  index  at  XII. 

(3)  Again  rotate  the  globe  until  the  heavenly  body  in 
question  is  brought  to  the  eastern  or  western  horizon,  accord- 
ing as  we  wish  to  find  the  time  of  rising  or  setting,  and  the 
hour  index  will  show  the  required  time. 

Similarly  the  time  at  which  a  heavenly  body  culminates 
may  be  found. 

EXERCISES. 

(1)  Find  by  means  of  a  globe,  the  apparent  times  of  sunrise 
and  sunset  on  the  15th  April. 

(2)  Find  how  long  twilight  lasts  on  the  same  date. 

(3)  Find  the  length  of  the  day  at  Dublin  (lat.  53°  20')  on 
25th  November. 

(4)  At  what  hour  does  Sirius  cross  the  meridian  of  Dublin  on 
(1)  the  10th  August,  (2)  the  14th  December? 

(5)  At  a  place  whose  latitude  is  47°  the  bright  star  Arcturus 
rises  on  a  certain  date  at  8  p.m.     "What  is  the  date  ? 

(6)  In  what  part  of  the  heavens  should  we  expect  to  see  the 
bright  star  a  Lyrse  at  8  p.m.  on  the  15th  October,  at  Dublin  ? 


TERRESTRIAL  MERIDIAN  DRAWN.  233 

rf 

To  draw  a  Meridian  Line  on  the  Earth. 

On  a  fixed  horizontal  plane,  describe  with  a  compass 
several  concentric  circles.  At  the  common  centre  of  the 
circles,  erect  a  small  piece  of  straight  wire  at  right  angles  to 
the  horizontal  plane.  In  the  forenoon,  let  the  point  where 
the  shadow  of  the  top  of  the  wire  just  touches  any  one  of 
circles  be  marked,  and  also  the  point  where  the  shadow  just 
reaches  the  same  circle  again  in  the  afternoon.  If  a  radius 
be  now  drawn  bisecting  the  arc  between  these  points,  this 
radius  will  coincide  with  the  meridian.  Several  circles  are 
drawn  lest  a  cloud  over  the  sun  should  interfere  with  the 
observation. 


EXAMINATION  PAPERS 


AND 


MISCELLANEOUS  QUESTIONS. 


EXAMINATION  PAPERS. 


The  following  questions  from  1  to  100  have  been  set  to  third 
and  fourth  year  Students  in  the  University  of  Dublin,  each  set 
of  ten  questions  constituting  a  paper.  The  remainder  of  the 
questions  are  taken  from  the  Degree  Examination  Papers  set 
at  the  London  and  Royal  Universities: — 


i. 

1.  State  what  you  know   of   " double  stars,"    "new  stars," 
"periodical  stars,"  and  "nebulae." 

2.  State  clearly  the  reason  why  the  discs  of  the  sun  and  moon  ^^ 
appear  oval  when  near  the  horizon. 

3.  "  The  north  celestial  pole,  therefore,  will  be,  about  13,000 
years  hence,  nearly  49°  from  the  polar  star."     Give  a  clear  expla- 
nation of  this  statement. 

4.  What  is  "  annual  parallax  "  ? 

(a)  If  p  be  the  number  of  seconds  in  the  annual  parallax  of  a 
fixed  star,  show  that  the  time  taken  by  light  to  reach  us  from  this 

star  is,  approximately,  —  years. 

5.  Give  an  account  of  the  different  explanations  of  the  planetary 
motions  called,  respectively,  the  Copemican  system  and  Ptolemaic 
system ;  and  show  how  the  former  may  be  verified  in  the  case  of 
an  inferior  planet. 

6.  Assuming  Kepler's  laws,  prove  that  the  velocities  of  any 
two  planets  are  connected  with  their  distances  from  the  sun  by  the 
relation 

v  :  v'  ::  Jr'  :  Jr. 

7.  Show  how  to  find  the  moon's  sidereal  period.     How  is  the 
exact  synodic  period  determined  ? 

8.  "What  is  the  cause  of  a  lunar  eclipse  ?    Why  does  the  pheno- 
menon not  occur  at  every  full  moon  ?    How  are  the  lunar  ecliptic 
limits  found  ? 


[4]  EXAMINATION    PAPERS. 

f  9.  Describe  the  Meridian  Circle,  and  show  how  it  may  be  used 
to  find  (a]  the  declination,  and  (b)  the  right  ascension,  of  a  star. 

10.  Find  the  time  at  Vienna  (16°  20'  E.),  and  at  Chicago  (87° 
40' W.),  when  it  is  10  o'clock,  a.m.,  at  Dublin  (6°  15' W.). 

II. 

1 1 .  Show  how  a  degree  of  a  meridian  is  measured ;  and  assum- 
ing the  length  of  a  degree  to  be  69£  miles,  find  the  earth's  diameter 
in  miles. 

12.  Show  by  a  figure  the  effect  on  the  position  of  a  star  of  the 
refraction  of  light  by  the  atmosphere ;  and  prove  that  the  amount 
of  refraction  varies  approximately  as  the  tangent  of  the  star's  zenith 
distance. 

^>  13.  "What  is  the  cause  of  twilight? 

(a)  Does  twilight  ever  last  all  night  at  Paris  (lat.  48°  50')? 
Give  a  reason  for  your  answer. 

14.  A    circumpolar  star    crosses    the    meridian    at    altitudes 
10°  11' 17"  and  72o  15>  31//;  find  tlie  latitude  of  the  place,    and 
the  star's  polar  distance. 

15.  The  interval  between  eastern  and  western  quadratures  of 
Jupiter  is  175  days,  and  between  two  oppositions  400  days,  approxi- 
mately ;  find  the  annual  parallax  of  this  planet. 

16.  Show  how  the  height  of  a  lunar  mountain  may  be  obtained 
by  measuring  the  distance  from  the  boundary  of  light  and  darkness 
of  a  bright  spot  observed  in  the  umlluminated  part  of  the  moon's 
disc,  and  prove  the  approximate  formula — 

height  in  miles  =  537m*  cosecV, 

where  m  is  the  ratio  of  the  observed  distance  to  the  moon's  radius, 
and  e  the  moon's  elongation. 

17.  Prove  the  formula  for  finding  the  periodic  time  of  a  superior 
planet  by  means   of  the  earth's   periodic  time  and  the  observed 
interval  between  two  successive  oppositions. 

1 8.  Prove  that  more  than  half  the  disc  of  a  superior  planet  is 
always  seen,  and  that  the  planet  is  most  gibbous  in  quadrature. 

19.  Give  a  general  explanation  of  solar  and  lunar  eclipses. 

>  20.  Define  the  "  equation  of  time,"  and  state  from  what  causes 
it  arises.  "What  is  its  greatest  value  ?  How  many  times  in  the 
year  does  it  vanish,  and  at  what  dates  ? 


EXAMINATION   PAPERS.  [5] 

21.  The  apparent  zenith  distances  of  y  Draconis  at  lower  and 
upper  culminations  were  75°  3'  13"-2  and  1°  53'  18"-6  south ;  the 
amounts  of  refraction  in  the  two  observations  being  3' 41"- 9  and 
l"-9,  respectively.   Find  the  declination  of  the  star,  and  the  latitude 
of  the  place. 

22.  Explain  how  the  meridian  altitude  of  a  star  can  be  observed 
correctly  to  the  fraction  of  a  second. 

23.  Give  a  direct  explanation  of  the  effect  of  aberration  on  the 
positions  of  stars,  and  indicate  on  the  celestial  sphere  the  point 
towards  which  they  are  displaced. 

24.  Calculate  the   number    of  seconds    in    the  coefficient  of 
aberration. 

25.  Explain  the  nature  of  the  phenomena — (a)  precession  of  the 
equinoxes  ;  (1}  nutation.     By  what  observations  may  their  existence 
be  detected  ?    "What  are  their  periods  and  amounts  ? 

26.  Determine  an  expression  for  the  angle  which  the  breadth  of 
the  shadow,  cast  by  the  earth  at  the  distance  of  the  moon,  subtends 
at  the  earth,  in  terms  of  the  semidiameter  of  the  sun  and  the  hori- 
zontal parallaxes  of  the  sun  and  moon.     "When  is  this  a  maximum, 
and  when  a  minimum  ? 

27.  What  is  meant  by  the  term  "  radiant  point  of  a  meteoric 
shower ' '  ?    By  what  arguments  is  the  connexion  between  comets  and 
meteors  established  ? 

28.  Assuming  that  the  planetary  orbits   are  circles   with  the 
sun  in  their  common  centre,   find  an  expression  for  the  annual 
parallax  of  Jupiter  in  terms  of  the  synodic  period,  and  the  interval 
between  two  consecutive  quadratures. 

29.  Explain  the  Lunar  Method  of  finding  longitude  at  sea,  and 
point  out  its  disadvantages. 

30.  Examine  the  statement : — "  If  the  moon  had  moved  round 
the  earth  in  about  three  days,  the  longitude  would  have  been  as 
easily  found  as  the  latitude." 

IV. 

31. -Describe  a  transit  instrument,  and  state  the  nature  of  the 
errors  for  which  allowance  has  to  be  made. 

32.  How  would  you  determine  the  angle  subtended  by  the 
earth's  disc  at  the  moon  ? 


[6]  EXAMINATION    PAPERS. 

33.  Prove  the  following  rule  for  determining  from  the  true 
position  of  a  star  a-  its  apparent  position  a-'  due  to  aberration : — 
Let  S  be  the  sun,  and  E  a  point  on  the  ecliptic  90°  behind  S,  then 
<r'  lies  on  the  great  circle  vE  and  oV  =  20"*5  sin  a-JE. 

34.  Prove  Bradley's  formula  for  the  coefficient  of  refraction, 
and  state  accurately  what  observations  have  to  be  made  in  apply- 
ing it. 

35.  Define  the  terms  "lunation"  and  "  periodic  time  "  of  the 
moon ;  and  find  the  periodic  time,  being  given  that  a  lunation  is 
29-5305887  days. 

36.  Verify  the  following  statement : — "  At  the  end  of  19  years 
the  sun  and  moon  return  to  the  same  relative  position  with  regard 
to  the  fixed  stars,  and  the  full  moons  fall  again  on  the  same  days 
of  the  month,  and  only  one  hour  sooner." 

37.  Show  by  a  figure  the  relative  positions  of  the  Plough,  Pole 
*tar,  Arcturus,  Spica,  Capella. 

38.  Prove  that  when  a  transit  of  Yenus  is  about  to  take  place, 
Venus  and  the  sun  approach  each  other  at   the  rate  of  4"  per 
minute. 

39.  Describe  Foucault's  pendulum  experiment  for  the  latitude 
of  Dublin,  and  give  its  explanation. 

40.  How  would  you  propose  to  draw  to  scale  the  orbit  of  the 
earth  as  a  result  of  observation  ? 

v. 

41.  Describe  a  transit  instrument,  and  define  accurately  its  line 
of  collimation. 

42.  Explain  how  an  angle  can  be  measured  to  a  fraction  of  a 
second. 

43.  State  some  facts  which  have  been  accurately  observed  which 
can  only  be  explained  on  the  hypothesis  of  the  earth's  rotation  on 
its  axis. 

44.  Give  a  direct  explanation  of  the  aberration  of  light,  and 
calculate  the  constant. 

45.  Assuming  that    the    amount    of    refraction  varies  as  the 
tangent  of    the  zenith    distance,  indicate   how  the  coefficient  is 
determined. 

46.  In  the  "Nautical  Almanac"  the  declination  of  the  sun  is  given 
for  times  separated  by  intervals  of  three  hours.     Show  how  it  can 


EXAMINATION    PAPERS.  [7] 

be  determined  for  any  instant  whatever,  and  hence  deduce  the 
latitude  of  the  place  from  one  observation  of  the  sun. 

47.  Kepresent    on   a   diagram    the    relative    positions    of   the 
equator,   ecliptic,   and  horizon  at  sunset  on  the  evenings  of  the 
vernal  and  autumnal  equinoxes ;  and  for  the  latitude  of  Dublin  in 
each  case  calculate  the  angle  between  the  latter  pair  of  circles. 

48.  Give  the  meanings  of  the  terms  nutation  and  precession  of 
the  equinoxes.     How  was  their  existence  detected  ? 

49.  Prove  the  following  statement,  connecting  the  mean  time  at 
a  given  meridian  with  the  corresponding  sidereal  time  : — 

Sidereal  time  =  mean  time  +  mean  sun's  right  ascension. 
*:  50.  What  observations  are  necessary  in  order  to  determine  the 
periodic  time  of  the  moon  ?    Give  the  formula  which  is  required 
in  the  subsequent  calculation. 

VI. 

51.  Assuming   that  the   earth  is  spherical,  explain  how 
diameter  may  be  measured. 

52.  Explain,  by  means  of  a  diagram,  how  the  change  in  the 
sun's  declination  produces  the  succession  of  the  seasons. 

53.  Determine  the  limits  of  the  latitude  of  places  at  which 
twilight    lasts    all    night   long,    when  the  sun's   declination  is 
+ 10°  15'. 

54.  Assuming  the  horizontal  parallax  of  the  moon  to  be  -gV, 
and  her  apparent  diameter  to  be  1963",  find  the  moon's  diameter 
in  miles. 

55.  The  interval  between  the  inferior  conjunctions  of  Mercury 
is  115'8  days  ;  find  the  periodic  time  of  Mercury. 

56.  "Write  a  short  account  of  the  rising  and  setting  of  the  moon 
at  different  seasons  of  the  year. 

57.  Explain  how  the  meridian  of  a  place  may  be  found. 

58.  Give  an  account  of  the  lunar  method  of  finding  the  longi- 
tude at  sea,  and  the  objections  to  it. 

59.  How  are  the  retrograde  and  stationary  appearances  of  the 
planet  Venus  explained  ? 

60.  What  is  Bode's  law  of  the  distances  of  the  planets  ? 

VII. 

61.  How  would  you  make  use  of  a  celestial  globe  to  find  out 
what  stars  would  be  visible  in  a  given  place  at  a  known  date,  and 
at  a  given  hour  of  the  night  ? 


[8]  EXAMINATION   PAPERS. 

62.  Assuming  that  all  the  other  corrections  have  been  made, 
how  would  you  make  sure  that  the  great  circle  in  which  the  line 
of  collimation  of  a  transit  instrument  moves  coincides  with  the 
meridian  of  the  place  ? 

63.  An  altitude  of  a  star  is  observed  and  found  to  be  the  angle 
whose  sine  is  1%-;  calculate  the  true  position  of  the  star,  assuming 
the  amount  of  refraction  at  an  altitude  of  45°  to  be  58"-2. 

64.  At  mean  noon  on  a  given  date,  the  sidereal  time  was  14 
hours ;  what  will  be  the  sidereal  time  50  days  after,  at  mean  noon, 
in  the  same  place  ?    You  are  given  that  the  length  of  a  tropical 
year  is  365£  days. 

65.  Explain  the  different  causes  that  enable  us  to  see  somewhat 
more  than  exactly  half  the  surface  of  the  moon. 

66.  A  star  in  the  ecliptic  has   a  longitude  or  75°,  obtain  the 
change  in  the  position  of  the  star  owing  to  aberration,  when  the 
longitude  of  the  sun  is  135°,  assuming  the  constant  of  aberration  to 
be  20"-45. 

67.  Explain  a  method  of  finding  the  ratio  of  the  distances  of 
Venus  and  the  earth  from  the  sun. 

68.  Describe  accurately  how  the  latitude  can  be  found  at  sea. 

69.  In  connexion  with  the  satellites  of  Jupiter,  we  can  observe 
the  following :  transits  of  their  shadows  over  his  disc,  eclipses,  occul- 
tations  and  transits  of  the  satellites ;  explain  these  phenomena  by 
means  of  a  diagram. 

70.  Assuming  that  the  orbits  of  the  planets  are  circles  described 
with  uniform  velocity  in  the  same  plane,  prove  the  formula  for  the 
periodic  time  of  a  planet  when  the  time  between  two  conjunctions 
is  known. 

VIII. 

71.  Given  a  celestial  globe,  describe  how  you  could  use  it  to 
find  out  at  what  time,  approximately,  Regulus  will  culminate  on 
January  23,  1893,  in  Dublin. 

72.  Define  the  terms — right   ascension,    declination,    celestial 
longitude,  azimuth. 

73.  The  apparent  diameter  of  the  sun  when  least  is  31'  32",  and 
when  greatest  32'  3  6"*  4 ;    hence  calculate  the  eccentricity  of  the 
orbit  of  the  earth  round  the  sun. 

74.  Give  a  general  description  of  the  movements  of  the  moon, 
round  the  earth. 


EXAMINATION    PAPERS.  [~9~| 

75.  Draw  a  figure  to  show  the  apparent  relative  sizes  of  Venus 
and  the  portions  of  her   disc  which   appear   bright,  just  before 
inferior  conjunction,  when  at  her  greatest  brilliance,  when  gibbous 
End  when  at  superior  conjunction,  respectively. 

76.  Jupiter's  outer  satellite  is  at  a  distance  of  1,170,000  miles 
from  Jupiter,  and  takes  16  days  16£  hours  to  complete  one  revolu- 
tion round  him;  given  that  the  innermost  satellite  is  at  a  distance 
of  262,000  miles,  find  the  time  it  takes  to  revolve  round  Jupiter. 

77.  Prove  that  the  apparent  motion  of  Mars  is  retrograde  when 
we  are  closest  to  him,  and  direct  when  we  are  farthest  from  him. 

78.  Owing  to  the  aberration  of  light,  a  star  can  at  most  be 
displaced  from  its  true  position  through  an  angle  of  20"  3  •  hence 
calculate  the  velocity  of  light  assuming  that  the  velocity  of  the 
earth  is  19  miles  per  second. 

79.  Assuming  that,  on  a  certain  day  at  Greenwich,  the  right 
ascension  of  the  mean  sun  was  10  hours  at  12  o'clock,  find,  for  a  place 
whose  longitude  is  60°  west,  the  time  by  an  ordinary  clock  on  that 
same  day,  when  the  time  by  an  astronomical  clock  at  the  place 
was   14   hours.     You  may  assume  that  a  sidereal  day  contains 
23h  56m  4s  mean  time. 

80.  In  what  respects  are  the  motions  of  the  planets  strikingly 
similar,  and  strikingly  different  from  the  motions  of  such  comets  as 
Encke's,  Biela's,  and  Halley's  ? 

IX. 

81.  Explain  by  what  measurements  and  calculations  the  length 
of  the  earth's  diameter  is  obtained. 

82.  State  and  prove  the  law  of  atmospheric  refraction. 

The  horizontal  refraction  is  about  35'.     How  is  this  proved  ? 

83.  State  the  arguments  for  the  annual  motion  of  the  earth 
round  the  sun. 

84.  State  what  you  know  of  solar  spots,  and  the  inferences 
drawn  from  observations  made  on  them. 

85.  If  an  observer  could  reach  the  North  Pole,  and  remain 
there   from   tne  autumnal  equinox  to  the  vernal  equinox,  what 
appearance  of  (1)  sunlight,  and  (2)  moonlight  would  he  observe? 

86.  Calculate  the  moon's  sidereal  period,  assuming  the  synodic 
period  to  be  29-53  days.     How  is  the  latter  period  determined 
accurately  ? 


[10]  EXAMINATION    PAPERS. 

87.  The  retardation  of  rising  on  successive  nights  of  the  new 
moon  nearest  the  vernal  equinox  is  less  than  that  of  the  new  moon 
at  any  other  time  of  the  year.     Explain  this  phenomenon. 

88.  State  the  cause  of  a  solar  eclipse,  and  explain  under  what 
circumstances  it  is  (1)  total,  (2)  partial,  or  (3)  annular. 

89.  Assuming  the  velocity  of  the  planet  Mercury  to  be  30  miles 
per  second,  determine  the  velocity  of  Saturn  by  an  application  of 
Bode's  law. 

90.  Explain  accurately  the  methods  of  finding  longitude  at  sea 
(1)  by  chronometers,  and  (2)  by  observations  on  the  moon. 


x. 

91.  What  are  the  observations  by  which  the  distance  from  the 
earth  to  the  sun  is  ascertained  in  terms  of  the  standard  yard?    (The 
earth  may  be  supposed  a  perfect  sphere.) 

92.  What  further  observations  determine  the  distances  of  certain 
of  the  fixed  stars  ? 

93.  The  latitude  of  John  o'  Groat's  house  is  58°  59'  IS".     Find 
the  sun's  meridian  altitudes  at  that  place  on  midsummer  and  mid- 
winter days,  respectively. 

94.  Why  is  it  that  some  of  the  planets  are  seen  at  all  angular 
distances  from  the  sun,  others  only  when  they  are  within  a  certain 
angular  distance  from  the  sun  ? 

95.  How  is  an  astronomical  clock  regulated  ? 

96.  How  do  you  account  for  the  fact  that  some  of  the  great 
meteoric  displays  are  periodic  ? 

97.  Explain  how  the  fact  that  the  moon  always  turns  the  same 
face  towards  the  earth  enables  its  time  of  revolution  on  its  own 
axis  to  be  calculated. 

98.  Why  is  summer  longer  than  winter  ?    Is  it  the  case  for  the 
southern  hemisphere  ? 

99.  How  does  the  duration  of  twilight  at  a  given  place  alter 
with  the  season  of  the  year  ? 

100.  How  is  it  shown  that  the  elevation  of  a  star,  due  to  refrac- 
tion, varies  as  the  tangent  of  the  zenith  distance  ? 


[11] 


XI. 

MISCELLANEOUS   QUESTIONS. 

101.  State  the  law  of  Kepler  relating  to  the  periods  of  the 
planets,  and  deduce  it  from  the  law  of  gravitation  for  the  case  of 
circular  orbits. 

Calculate  the  periodic  time  of  a  meteorite  describing  a  circular 
orbit  round  the  sun  close  to  its  surface.  (B.A.  Lond.) 

102.  How  long  does  the  sun  take  to  rise  at  a  point  on  the 
Equator  on  March  21st?     Is  this  interval  greater  or  less  than  at 
other  times  and  places  ?     How  much  would  it  be  altered  for  the 
case   of   an  observer   on  the  deck   of  a  ship  sailing  due  east  at 
10  miles  an  hour?     [The  diameter  of  the  sun  may  be  taken  as 
half  a  degree.]     (B.SC.  Lond.) 

103.  How  is  the  declination  of  a  heavenly  body  obtained  by 
observation  ? 

The  declination  of  the  moon's  centre  is  observed  at  the  same 
instant  at  two  observatories  in  different  latitudes ;  show  that  the 
results  will  differ  by  an  angle  of  the  same  order  of  magnitude  as  a 
degree,  and  state  precisely  the  meaning  of  the  moon's  declination 
as  tabulated  in  the  "  Nautical  Almanac."  (B.SC.  Lond.) 

104.  The  constellation  of  the  Southern  Cross  is  in  right  ascen- 
sion 12°,  and  North  Polar  distance  152°;  find  the  most  northerly 
latitude  at  which  it  is  ever  visible,  and  the  time  of  year  at  which  it 
may  be  seen  at  such  a  place.     (B.SC.  Lond.) 

105.  Give    an   exact    definition   of   longitude   on   the   earth's 
surface,  and  state  the  principle  by  means  of  which  longitudes 
are  ascertained. 

Find  the  ratio  of  the  lengths  of  a  degree  of  longitude  at  the 
equator  and  in  latitude  A,  supposing  the  earth  to  be  a  sphere. 
(B.A.  Lond.) 

106.  Explain  the  cause  of  the  seasons.     What  would  be  their 
character  if  the  earth's  axis  were  nearly  at  right  angles  to  the 
ecliptic.     (B.A.  Lond.) 

107.  How  is  a  star's  parallax  determined  ?  What  is  the  distance 
of  a  star  which  shows  a  parallax  of  0"'5,  correct  to  the  number  of 
figures  that  would  be  reliable  in  practice  ?     (B.A.  Lond.) 


[12]  MISCELLANEOUS   QUESTIONS. 

108.  Given  that  one  lunation  is  29*5306  days,  and  the  period 
of  the  synodic  revolution  of  the  moon's  node  is  346 -66  days;  prove, 
with  full  explanations,    that  eclipses  of  the  sun  and  moon  will 
repeat  themselves  after  an  interval  of  ahout  18  years  10  days. 
(B.SC.  Lond.) 

109.  Explain  the  influence  of  the  aherration  of  light  on  astro- 
nomical observations.     Should  observations  of  the  stars  he  corrected 
for  aherration  due  to  the  motion  of  the  solar  system  through  space  ? 
(B.SC.  Lond.) 

110.  The  eccentricity  of  the  earth's  orbit  is  -§V;  find  what  per- 
centage of  the  coeflicient  of  aberration  is  in  consequence  variable 
throughout  the  year.     (B.SC.  Lond.) 

111.  Why  is  it  that  the  time  of  sunrise,  as  observed  by  an 
ordinary  clock,  becomes  later  for  some  days  after  the  shortest 
day?     (B.A.,  R.TJ.I.) 

112.  From  what  observations  has  the  precession  of  the  equi- 
noxes been  determined  ?     "What  is  its  effect  on  the  right  ascension, 
declination,  latitude,  and  longitude  of  a  star.     (B.A.,  E.TJ.I.) 

113.  How  would  you  find  the  sun's  declination  when  it  rises 
at  the  N.E.  point  of  the  horizon  ?    (B.A.,  B.TLI.) 

114.  Knowing  the  periodic  time  of  Venus  and  the  earth,  how, 
from  observing  the  interval  between  the    greatest   eastern  and 
western  elongations  of  the  planet,  can  its  distance  from  the  sun  be 
compared  with  the  radius  of  the  earth's  orbit?     (B.A.,  R.TJ.I.) 

115.  Given  the  maximum  elongation  of  an  inferior  planet,  cal- 
culate its  periodic  time.     What  is  the  synodic  period  of  a  planet  ? 
(B.A.,  B.TT.I.) 

116.  Show  that  the  moon,  moving  under  the  attraction  of  the 
earth  and  the  sun,  is  always  subject  to  a  component  force  tending 
to  the  sun,  and  therefore  that  its  orbit  round  the  sun  is  concave  to 
that  body.     [The  radius  of  the  moon's  orbit  may  be  taken  as  -j-iroth 
of  the  radius  of  the  earth's  orbit,  and  the  lengths  of  the  month  and 
year  may  be  assumed.]     (B.SC.  Lond.) 

117.  Richer  discovered,  in  the  17th  century,  that  a  pendulum- 
clock,  regulated  at  Paris,  lost  2  minutes  28  seconds  daily  when 
taken  to  Cayenne  ;  deduce  the  conclusion  that  the  value  of  gravity 
is  smaller  at  Cayenne  than  at  Paris  by  "11  of  a  foot-second  unit 
approximately.     (B.SC.  Lond.) 

118.  How   are   the   masses   of    the   planets   determined   from 
observations  of  the  periodic  times  of  their  satellites  ?     (B.SC.  Lond.) 


MISCELLANEOUS    QUESTIONS.  [13] 

119.  Explain  how,  by  observation  of  the  transits  of  a  circum- 
polar  star  across  the  meridian,  to  determine — (1)  the  declination  of 
the  star ;  (2)  the  latitude  of  the  observatory. 

An  error  of  one  second  in  the  latitude  corresponds  to  an  error 
of  how  many  feet  in  position  on  the  earth's  surface  ?  (B.SC.  Lond.) 

120.  How  is  the  rate  of  error  of  the  astronomical  clock  in  an 
observatory  exactly  determined?     (B.A.  Lond.) 

121.  What  is  the  equation  of  time?    How  does  its  magnitude 
vary  throughout  the  year  ? 

On  November  1st  the  sun  rises  at  6h  56m  and  sets  at  4h  31m  mean 
time ;  find  the  equation  of  time  for  that  day,  (B.SC.  Lond.) 

122.  How  might  the  length  of  a  degree  of  longitude  be  directly 
measured  ?     How  does  it  vary  in  different  latitudes  ? 

Truro  is  marked  on  a  map  as  being  20  min.  32  sec.  slow  by 
Greenwich  time,  and  Norwich  as  5  min.  8  sec.  fast ;  what  is  the 
difference  of  their  longitudes  in  degrees?  (B.SC.  Lond.) 

123.  Explain  how  the  moon's  distance  has  been  determined. 
(B.SC.  Lond.) 

124.  Describe  and  explain  the  effect  of  the  aberration  of  light 
on  the  apparent  position  of  the  stars.     (B.SC.  Lond.) 

125.  Show  that  a  swarm  of  meteors  passing  through  the  earth's 
atmosphere  in  parallel  lines  appears  to  radiate  from  a  point ;  and 
show  how,  by  taking  account  of  the  direction  of  this  point  and  the 
motion  of  the  earth  and  the  velocity  of  the  meteors,  to  determine 
the  direction  of  their  actual  motion  through  space.    (B.SC.  Lond.) 

126.  If  the  apparent  meridian  altitudes  of  a  circumpolar  star 
are  45°  and  60°,  find  the  latitude  of  the  place  and  the  declination  of 
the  star,  the  coefficient  of  refraction  being  58" '2. 

127.  What  conditions  would  have  to  be  taken  into  account  in 
calculating  when  Yenus  is  brightest  ?     Give  her  position  approxi- 
mately, and  describe  her  appearance  when  she  is  brightest. 

128.  When  the  astronomical  time  at  Dunsink  (longitude  25m22"W.) 
was  5h  10m  16s  on  September  1st,  1893,  what  was  the  mean  time? 
You  are  given  that  the  right  ascension  of  the  mean  sun  at  mean 
noon  at  Greenwich  on  September  1st,  1893,  was  10h  43m  29s,  and 
that  the  earth  rotates  on  its  axis  in  23h  56m  4s. 

129.  If  a  star  is  situated  on  the  ecliptic,  show  that  its  parallax 
is  nothing  when  its  aberration  is  a  maximum,  and  vice  versa  when 
its  aberration  is  nothing,  its  parallax  is  a  maximum. 


[14]  MISCELLANEOUS    QUESTIONS. 

130.  The  right  ascension  of  Sirius  is  6h  38m,  and  the  south 
declination  is  16°  30'.     In  what  month  would  you  expect  it  to  be 
due  south  about  6  o'clock  in  the  evening  ?    Would  you  expect  it  to 
be  visible  in  June  at  any  time  in  the  British  Isles  ?    State  generally 
in  what  portions  of  the  world  you  would  expect  it  to  be  visible  in 
that  month. 

131.  How  would  you  calculate  the  coefficient  of  refraction  by 
observations  on  one  circumpolar  star  whose  declination  is  known  ? 

132.  Calculate  the  average  retardation  of  the  moon  in  rising, 
and  show  by  a  figure,  for  the  latitude  of  Dublin,  when  the  actual 
retardation  is  greatest  and  least. 

133.  Explain  how  it  is  that  Venus  appears  both  as  an  evening 
and  a  morning  star.     Jupiter  and  Venus  are  evening  stars,  and 
stationary ;  find  which  way  they  will  begin  to  move.     (E.TJ.I.) 

134.  Does  the  time  given  by  a  sun-dial  agree  with  that  given  by 
an  ordinary  clock  ?     Explain  your  answer,  and  examine  when  that 
part  of  the  equation  of  time  arising  from  the  eccentricity  of  the 
earth's  orbit  is  positive.     (E.TJ.I.) 

1 35.  Prove  that  the  displacement  of  a  heavenly  body  by  refraction 
is  approximately  proportional  to  the  tangent  of  the  zenith  distance 
for  moderate  zenith  distances ;  and  show  how  the  index  of  refraction 
of  air  at  the  earth's  surface  might  be  determined  by  observing  the 
two  meridian  altitudes  of  a  circumpolar  star  whose  declination  is 
known  ?    (B.TJ.I.) 

136.  In  an  observatory,  how  would  you  proceed  to  determine 
the  right  ascension  of  a  fixed  star,  the  index  error  of  the  sidereal 
clock  and  the  right  ascensions  of  all  other  stars  being  supposed 
known.     (B.A.,  E.U.I.) 

137.  To  what  is  the  equation  of  time  due  ?     Trace  approximately 
its  value  throughout  the  yeaj ;  and  state  whether  it  remains  constant 
for  the  same  day  in  successive  years.     (B.A.,  E.TJ.I.) 

138.  The  declination  of  a  given  star  is  known.    Show  how, 
from  observations  of  it,  you  would  deduce  your  latitude.    (B.A., 
E.TJ.I.) 

139.  Describe  the  phenomenon  known  as  aberration.      Show 
that  from  it  and  Foucault's  determination  of  the  velocity  of  light, 
the  earth's  distance  from  the  sun  may  be  deduced.     (B.A.,  E.TJ.I.) 

140.  Assuming  that  the  planets  move  round  the  sun  in  circles, 
use  Kepler's  third  law  in  deducing  the  law  of  inverse  squares  as 
regards  gravitation.     (B.A.,  E.TJ.I.) 


MISCELLANEOUS    QUESTIONS.  [15] 

141.  Determine  the  lowest  latitude  at  which  twilight  lasts  all 
night.     Why  is  it  that  twilight  does  not  last  so  long  in  the  tropical 
regions  as  elsewhere  ?     "When  is  its  duration  shortest  ?     (B.TT.I.) 

142.  Explain  how  the  latitude  of  a  place  can  be  determined  by 
observing  a  known  star  when  crossing  the  meridian.     At  a  place  in 
latitude  50°  K,  what  are  the  limits  of  position  of  those  stars  which 
remain  always  above  the  horizon?     (R.TJ.I.) 

143.  Represent,  on  one  diagram,  the  celestial  equator,  horizon, 
ecliptic ;  the  latitude,  longitude,  declination,  right  ascension,  hour 
angle,  and  azimuth  of  a  star. 

144.  Name  the  principal  stars  of  tl^  first  magnitude  visible  to 
inhabitants  of  Dublin,  and  point   out  how  you  would  recognise 
them. 

145.  Suppose  you  were  asked  to  determine  the  errors  in  a  transit 
circle,  describe,  in  proper  order,  how  you  would  proceed.     Explain 
also  what  is  meant  by  the  polar  and  zenith  points. 

146.  How  are  the  coefficient  of  refraction  at  and  the  latitude  of 
a  place  determined  by  the  same  set  of  observations  ? 

147.  Explain  how  you  would  determine  the  altitude  of  the  sun 
by  means  of  an  artificial  horizon. 

148.  About  what  time  on  the  21st  March  is  the  first  point  of 
Aries  on  the  horizon  ?     Within  what  limits  does  the  true  time  lie  ? 

149.  Describe  a  sundH 

150.  It  sometimes  happens  that  three  eclipses  occur  within  one 
month.     Explain. 

151.  State  fully  the  arguments  from  which  the  connexion  between 
meteoric  showers  and  comets  has  been  inferred. 

152.  How  are  the  local  time  and  longitude  at  sea  determined? 

153.  How  is  the  angle  which  the  earth  subtends  at  the  moon 
found? 

154.  Describe  the  position  and  motion  of  the  moon's  orbit  in 
space,  and  explain  why  eclipses  recur. 

155.  The  apparent  zenith  distances  of  a  star  at  upper  and  lower 
culminations  were  75°  3'  13",  and  1°  53'  19"  south;  the  amounts 
of  refraction  were  3'  42",  and  2",  respectively.     Find  the  latitude 
of  the  place.     Draw  a  figure  to  represent  the  position  of  this  star 
at  its  upper  and  lower  transits,  and  insert  the  numbers. 


[16]  MISCELLANEOUS     QUESTIONS. 

156.  What  is  the  theory  of  meteoric  showers? 

157.  Indicate  how  you  would  verify  the  following  statement: — 
"At  the  end  of  19  years  the  sun  and  moon  return  to  the  same 
position  with  regard  to  the  fixed  stars,  and  the  full  moon's  fall  again 
on  the  same  days  of  the  month,  and  only  an  hour  sooner." 

1  year  =  365-25  days.     1  lunation  =  29-53  days. 

158.  How  is  the  radius  of  the  earth  determined  in  miles? 

159.  You  are  in  an  observatory,   and  are  in  possession  of  a 
"  Nautical  Almanac  " ;   explain  how  you  would  test  the  accuracy  of 
your  watch. 

160.  How  would  you  oraw  a  meridian  line  at  any  Dlace  on  the 
earth? 


INDEX. 


N.B.— The  Numbers  refer  to  the  pages. 


ABERRATION  of  light,  144  ;  effect  on 
position  of  stars,  145 ;  when  a 
maximum,  146. 

Adams,  discovery  of  Neptune,  102. 

Algol,  cycle  of  changes  of,  219. 

Altitude,  13. 

—  of  Pole,  21. 

Altitude  and  azimuth  instrument,  50. 

Andromeda,  nebula  in,  225. 

Andromedes,  the,  107. 

Angles,  measurement  of,  40,  51. 

Aphelion,  107. 

Apparent  time,  184. 

Areas,  Kepler's  Second  Law,  88. 

Ascension,  right,  13. 

Asteroids,  100. 

Atmosphere  of  moon,  168 ;  of  planets^ 
224. 

Axis  of  the  earth,  20. 

Azimuth,  13. 

error  of,  37. 


Ball,  Sir  B.  S.,  72,  217,  226. 

Beer    and    Madler,   heights  of    lunar 

mountains,  165. 
Bessel,    annual    parallax    determined, 

129. 

Biela's  comet,  107. 
Bode's  Law,  91. 
Bradley,  aberration  of  light,  148. 

coefficient  of  refraction,  59. 

nutation,  143. 


Calendar,  Julian  and  Gregorian,  197. 
Cavendish,  density  of  the  earth,  228. 


CelestiA  globe,  note  on,  231. 
CentauW0,  131 ;  mass  of,  230. 
Chromosphere,  223. 
Circle,    declination,     15  ;    great    and 

small,  2  ;  meridian  or  transit,  39. 

40. 

Clairaut,  on  Halley's  Comet,  104. 
Clock,  Astronomical,  33  ;  rate  of,  45 ; 

regulation  of,  45. 
Colatitude,  44. 
Collimation,  error  of,  36. 

line  of,  35. 

Colours,  prismatic,  220. 

Coloured  stars,  218. 

Comets,     103  ;      distinguished    from 

planets,    104  ;     connexion    with 

meteors,  107. 
Conjunctions,  inferior   and  superior, 

79. 

Copernicus,  system  of,  8. 
Corona,  sun's,  223. 
Cycle,  metonic,  155. 


Day,  apparent  solar,  184  ;  mean  solar, 

184 ;  sidereal,  33. 
Declination,  13  ;  found  with  meridian 

circle,  44. 

Degree  of  meridian  measured,  22. 
Delisle's  method    of   obtaining  sun's 

parallax,  121. 
Density  of  the  earth,  227. 
Diameters :  of  earth,  23 ;  of  the  moon, 

the  planets,  or  the  sun,  125. 
Distance  of  the  moon,  120,  150 ;  the 

sun,  62,  122;  stars,  128. 
Double  stars,  218. 


[18] 


INDEX. 


Earth,  annual  motion  of,  65  ;  axis  of, 
20 ;     density    of,    227  ;    diameter 
of,  23  ;  diurnal  motion  of,  25-31; 
eccentricity  of  orbit,  110;  shape 
of,  22. 

Eccentricity  of  an  elliptic  orbit,  110. 
Eclipses,  frequency  of,   180-3;    phe- 
nomena seen  during  lunar  eclipses* 
170  ;  satellites  of  Jupiter  eclipsed, 

-  lunar,  169  ;  solar,  173  J 
lunar  ecliptic  limits,  178 ;  solar 
ecliptic  limits,  179. 

Ecliptic,  11  ;  change  in  obK^ty  of, 
142;  obliquity  of,  11,  13W 

Ellipse,  89. 

Elongation,  80. 

Encke's  Comet,  104. 

Equation  of  time,  185-9  ;  curve 
showing  variations  in,  189  5 
vanishes  four  times  yearly,  188. 

Equator,  celestial,  10. 

terrestrial,  20  ;  celestial 

sphere  seen  from,  24. 

Equatorial  telescope,  50. 

Equinoxes,  12  ;  causes  of  the  pre- 
cession of  the,  139  ;  period  of  the, 
138  ;  precession  of  the,  138. 

Eratosthenes,  measurement  of  earth,  23- 

Error,  collimation,  36 ;  deviation,  37  ; 
level,  37  ;  eccentricity,  41. 

Flamsteed,  right  ascension  of  a  sta 

determined,  134. 
Foucault,  rotation  of  earth  proved  by 

pendulum  experiment,  30. 
Fraunhofer,  lines  in  solar  suectrum, 

221. 

Oolden  numbers,  155. 
Gravitation,  law  of,  90. 
Gregorian  Calendar,  198. 

Hadley's  sextant,  202. 

Halley,   transit    of  Venus  and  sun's 

parallax,  123. 
Halley 's  comet,  104. 
Harvest  moon,  161. 
Herschel,  102,  216. 
Hicetas,  9. 


Horizon,  the,  3,  9. 

Hour  angle,  15;  hour  circles,  15. 

Jupiter,  100;  satellites  of,  101. 

Kepler's  laws,  88. 

Kirchoff,  solar  spectrum,  223. 

Latitude,  celestial,  14. 

Latitude,  terrestrial,  20  ;  at  sea,  how 

found,  206-9. 
Leonids,  107. 

Leverrier,  discovery  of  Neptune,  102. 
Librations  of  the  moon,  158. 
Light,  aberration  of,  144  ;  composition 

of,  220  ;  velocity  of,  143. 
Local  time,  how  found  at  sea,  209. 
Longitude,  celestial,  14. 
Longitude,  terrestrial,  20;    found  by 

chronometers,     210  ;     by    lunar 

distances,  211. 

Magnitudes  of  stars,  214. 

Mars,  99  ;  satellites  of,  100. 

Maskelyne.  density  of  the  earth,  227. 

Mass,  of  the  earth,  227  ;  of  the  planets, 
229  ;  of  the  sun,  229. 

Mean  time,  33,  184. 

Mercury,  96  ;  seldom  seen,  81 ;  transits 
of,  98. 

Meridian,  10,  20;  altitude,  17,  43; 
circle,  39:  how  determined,  35-38  ; 
also  Note  on  Celestial  Globe,  231. 

Meteors,  connexion  with  comets,  107. 
showers  of,  106—7. 

Metonic  cycle,  155. 

Micrometer,  parallel  wire,  51 ;  screw, 
41,  51. 

Milky  Way,  216. 

Mira,  a  type  of  variable  stars,  219. 

Moon,  6 ;  distance,  150 ;  earth-shine, 
156  ;  harvest  moon,  161  ;  incli- 
nation of  path  to  ecliptic,  149  ; 
librations,  158 ;  lunar  mountains, 
165  ;  mean  angular  velocity,  150  ; 
periodic  time  found,  153;  phases, 
151 ;  physical  state,  167  ;  retar- 
dation, 160 ;  revolution  of  nodes, 
163  ;  synodic  period  found,  153. 


INDEX. 


[19] 


Nadir,  10  ;  to  point  telescope  at,  42. 
Nebulae,  216  ;  spectra  of,  225. 
Neptune,  discovery  of,  102  ;  exception 

to  Bode's  law,  92. 

Newton,  law  of  universal  gravitation,  90. 
Nodes,  moon's,  1G3, 180 ;  a  planet's,  80. 
Nutation,  141 ;    cause  of,  142 ;  period 
of,  142. 

Obliquity  of  ecliptic,  11 ;  how  found, 

137. 
Occupation  of  Jupiter's  satellites,  101 ; 

of  a  star  by  moon,  168,  213. 
Opposition,  79,  152. 

Parallax,  annual,  126  ;  Jupiter's,  131 ; 
star's,  129. 

diurnal,  114;    law  of,   117; 

moon's,  119;  sun's,  121   3. 

Pendulum  experiment  to  prove  earth'8 
rotation,  28-31. 

Penumbra,  170. 

Perihelion,  107. 

Periodic  time,  87,  152  ;  how  found 
for  moon,  153  ;  for  planets,  87. 

Perseids,  107. 

Phases  of  inferior  planets,  83  ;  of  the 
moon,  151;  of  superior-planets,  84. 

Phenomena  due  to  change  of  place  on 
earth,  23-5. 

Phobos,  extraordinary  period  of  revo- 
lution round  Mars,  100. 

Photosphere,  223. 

Planets,  7;  brightness  of,  85;  dia- 
meters, 124  ;  direct  and  retrograde 
motions,  92 ;  distinguished  from 
comets,  104  ;  inferior  and  supe- 
rior, 79  ;  interior  and  exterior,  79  ; 
masses  of,  229  ;  nodes,  80  5 
periodic  time  determined,  87  • 
phases  of,  83 ;  rotations  of,  95 ; 
stationary  points,  92  ;  synodic 
time,  87. 

Pleiades,  the,  216. 

Pole,  celestial,  4  ;  altitude  of,  21  ; 
movement  of  (precession),  138. 

Pole,  terrestrial,  20  ;  phenomena  of 
day  and  night  at,  24. 

Precession  of  the  equinoxes,  138 ; 
causes  of  precession,  139. 


Proper  motions  of  the  stars,  217. 
Ptolemaic  system,  7. 

Radiant  point  for  meteoric  showers, 
106. 

Refraction,  atmospheric,  53 ;  coeffi- 
cient of,  58  ;  effect  on  sun's  disc, 
60 ;  law  of,  66. 

Right  ascension,  14,  45 ;  of  a  Aquilse 
determined  by  Flamsteed,  134. 

Rotation  of  the  earth,  25 ;  of  the  moon, 
157  ;  of  the  planets,  95  ;  of  the 
SMI  71. 

Saros  of  the  Chaldeans,  182. 

Satellites,  Mars',  100  ;  Jupiter's,  101 ; 
Saturn's,  102 ;  Neptune's,  103. 

Saturn,  101 ;     rings  of,  102. 

Schiaparelli,  rotation  of  the  inferior 
planets,  95. 

Screw,  micrometer,  41,  51. 

Seasons,  explanation  of,  66. 

Sextant,  Hadley's,  202. 

Sidereal  day,  33  ;  time,  34,  192; 
year,  197. 

Signs  of  the  Zodiac,  12. 

Solar  system,  8,  78. 

Solstices,  16. 

Spectroscope,  construction  of,  220. 

Spectrum  analysis,  221 ;  applied  to 
the  moon  and  planets,  224 ;  to 
nebulae,  225  ;  to  the  fixed  stars, 
224  ;  to  the  sun,  221. 

Sphere  defined,  1  ;  celestial,  3  ;  oblique, 
parallel,  and  right,  23-5. 

Stars,  binary,  218;  circumpolar,  4; 
diurnal  motion  of,  3  ;  double, 
218;  magnitudes,  214;  number 
visible  to  naked  eye,  215  ;  paral- 
lax and  distance  of,  129 ;  proper 
motions  of,  217 ;  spectra  of,  224. 

Star  clusters,  216. 

Stationary  points  of  planets,  92. 

Sun,  atmosphere  of,  223  ;  diameter  of, 
62  ;  distance  of,  62, 117 ;  eclipses 
of,  173 ;  heat  from,  68 ;  mass  of, 
229  ;  motion  through  space,  218  ; 
parallax  determined,  121,  123; 
rotation  of,  71 ;  spectrum  of,  221 ; 
spots  on  surface  of,  71. 


[20] 


INDEX. 


Synodic  period,  87,  152;  found  for 
moon,  153. 

Time,  apparent,  184  ;  mean,  33, 184 ; 
sidereal,     33  ;      conversion    of, 
192-5. 

Transit  instrument,  34;  adjustment 
of,  to  meridian,  36  ;  method  of 
using,  38. 

Transit  circle,  see  Meridian  circle. 

Transits,  of  Mercury,  98  ;  of  Venus, 
97  ;  of  satellites  of  Jupiter,  101. 

Tropics,  16,  20. 

Twilight,  72  ;  duration  of  !™;  last- 
ing all  night,  76. 


Uranus,  102. 

Velocity,  of  the  earth,  143  ;  of  light, 
143  ;  of  the  planets,  ratios  of, 
96. 

Venus,  greatest  brilliancy,  85  ;  morn- 
ing or  evening  star,  81  ;  phases 
of,  83  ;  rotation  of,  95  ;  transits 
of,  97. 

Vertical,  circles,  10  ;  prime,  10. 

Tear,  tropical,  sidereal,  and  civil, 
197. 

Zenith,  10  ;  distance,  13. 
Zodiac,  the,  12  ;  signs  of,  12. 


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